How to organize a playday

I'll share my experience and the system that has sort of evolved over time- take what you will or feel free to charge off in your own direction with it. You may need to make some adjustments because in reality, it's your relationship with your local track and accommodating your proposed playdays to how they run their program that is the key to making this work. 
I. Contacting the track:  If you have an established relationship with a track, this could mean a simple phone call or email. If you don't, then you should attend one of their races, find out who the board members are and see if you can find one advocate on the board. Plan on attending one of their board meetings to pitch the idea to the entire board. Remember to entice them with the idea that you folks are more than glad to help boost the treasury of the club by paying the non-member practice fee in order to have a place to run. If they are unfamiliar with vintage karts, explain that these are vintage racing karts, not yard karts, and that your purpose is to have a place for a group of local enthusuasts who have restored or rebuilt karts from the earlier era of the sport to be able to fire them up  and run them at speed. Stress that you are not looking to hold a race, or have a free-for-all where anyone can bring out a beat-up craigslist kart with a worn out briggs and stratton on it. You are just looking for an opportunity to have open track time for vintage kart enthusiasts to be able to drive and enjoy the vintage racing karts they've worked hard to bring back to life in a non-competitive environment.
Also stress that just because the karts are vintage, no one is expecting to go out onto the track with a 40 year old helmet. A minimum of current helmets, neck collars, abrasion resistant jackets, long pants and closed toe shoes are part of the package. I have the luxury of living in an area where there are six kart tracks within a two hour radius. Of the six, at the time of the first playday, three weren't interested in having vintage karts when I contacted them and of the three remaining, only one was really enthusiastic about having us there. Hopefully you have multiple options. If you only have one track and they seem reluctant, see if you can entice a board member or two by bringing your vintage kart to the board meeting, or even offering to meet them at the track and let them take it for a spin. If they can see the allure, and the passion of those involved and realize that you're not talking about banged up yard karts, they should be more apt to listen.
II. Explaining/negotiating your program: When I did the first playdays, I was lucky that the meets I was involved with previously had provided a basis for what didn't work- and the number one thing is that you just don't mix modern karts and vintage karts out on the track. This is for a number of reasons. Foremost, the different lines and handling characteristics between vintage (and I'm talking 1975 and before) karts and modern karts do not make for good trackmates. While we may be clipping the apex knowing that we'll drift to the outside corner at the exit, a modern karter who often times has never even seen a vintage kart, does not know this and doesn't know what to anticipate.
At best, the vintage karts will be mobile chicanes. At worst, the mixing causes an on track incident and that ensures that you never get invited back to hold another playday. You also might have folks in your group who have early low-powered karts, which are really fun to watch and drive and are all part of the hobby- but that makes the problem of running with modern guys even more hazardous. We also don't have any bodywork or nerf bars, so I think vintage karters drive with that in mind. Only vintage karters understand how difficult it is to restore a vintage kart with the period goodies, and I think that adds a layer of respect and caution when you are out on the track with other vintage karts. I don't think that modern racers understand that you can't just order up a replacement part from Comet Kart Sales if something happens, especially when the mentality is "that's racin!" 
To be fair, the modern karts that are out on the track on a practice day are also usually there for a purpose. They are contesting a season for points and a championship. So they are there to break in a motor, or practice for an upcoming race- both things have a fair degree of more seriousness and required concentration than the vintage karting playday approach does.
So the main thing is to explain this in order to get the track to agree to give vintage a separate session. What has typically worked for us is we alternate 20 minute sessions with the regular club members. That said, at times when there was a low turnout on our end, and there would be a session or two where only one or two or sometimes no one was on the track while modern guys were more than ready to go out, we'd scale it back to 20 minutes at the top of the hour, giving the regular club members the remaining 40 minutes. It's a little embarrassing on our part to force guys who have work to do with their racing program to wait for their session when we're not going to utilize our sessions to the fullest. Being good guests is vital to the success of the playday and getting the opportunity to return. You'll have to come up with an agreement that works for your situation.  Regardless, you'll find that you'll still get plenty of running time by the end of the day. Even if you only get a 20 minute session each hour, if you are there all day you'll have 2 hours and 40 minutes of solid track time.  That's a lot of driving!
If you establish a working relationship with a club, you may see that things evolve over time. We've gotten to the point with ours where the club posts on their website that vintage karts are coming on such-and-such a date, and modern kart traffic is usually pretty light, and those that come out and run, break in their motors or test what they need to test in the morning and leave. So we usually end up with the track pretty much to ourselves from about noon on.
The other important thing to do is check out the club's race schedule and propose a Saturday that is farthest away from their race weekend. Out here the tracks we typically go to have one race a month. So I try to pick the weekend after their race, or if that doesn't work, two Saturdays before the next race. That way you don't mess up anyone in the club who needs the track time to prepare for a race the next weekend. Crossover dates work well too. With so many clubs in the area, there are weekends where the club goes and has a combined race with another club down the road. We pretty much end up with the track to ourselves on those days. Proposing dates in the fall, after the club's racing season has ended works well too.  
III. Contacting folks and getting the word out: Anytime you go to organize a playday, it's always wise to get some firm committments from your buddies before you approach the track, just so you're putting up a good faith effort and not wasting the club's time. Don't be discouraged. You would be surprised at how few vintage karters it actually takes in order to constitute "not wasting the club's time." We've had six guys before and it was worthwhile for both parties. I guess it's because at club races, you'll find they run classes that might only have three karts in them. If you can get 10-12 guys, that's a boon. If it grows from there, that's a big deal at a club's practice day.
Counter to the drill when running a meet where you have a break even point so you're hustling to get as many people to attend a meet as possible, there are no upfront costs to putting on a shared playday (remember that win-win deal?) so it's important to get like-minded folks together.  Those first few were just a couple emails and phone calls to my friends who I absolutely knew were on the same page as me. It has now eveolved to where I have an email group assembled and I'll throw a general announcement up on the various vintage karting boards. If anyone wants me to add them to my email group, I can do that and I just send out one email that goes to everyone who has asked me to add them.
In time, I realized the byproduct of the playday setup- open track time, no classes, no heats, no flags- was that the guys who are only interested in racing glory stay away. It's not worth it to them to load all the stuff up and drive to a playday if there is no thrill of competition. That allieviates a lot of the conflict that exists elsewhere in the hobby. That said, the host club is trusting you to keep it together because they don't know the hobby and that involves a lot of self-policing. You want to make sure you have folks that are all on the same page- especially when you are trying to get your playdays established. Without trying to sound like a 1950's civics class film, remember that you are the ambassadors of the vintage karting hobby. It doesn't matter if no one you're in contact with "converts" to vintage karting, it is important that you earn the respect of the club members though. When inviting folks, stick to your program and be upfront about what it is, and what it isn't- even if that means a couple folks decide not to come out.
IV. Giving it some personality: After you do a few playdays, you can always play around a little with the format and add some things. At most of the playdays here, we have a potluck BBQ, which started with simple brats and dogs, but has grown into an absolute feast - to the point where I'm starting to think we might be a BBQ group that also plays with vintage karts. At times we've had an informal swap meet. Out here the November Turkey Trot playday (always the Saturday after Thanksgiving!) has grown to where we're going to try to formalize the swap meet portion and see how that goes, and at some point we might do an informal show to get some of the area karters out who have show karts but don't run them. A simple kind of "people's choice" deal with the show happening while everyone is taking the BBQ break might be a fun deal. But there are ways to do those kinds of things without turning it into a big production where someone needs to put money up front and then worry about at least breaking even. Get the ball rolling and just see where it takes you.