Lemons/Chump team challenges
#1 Getting someone to give up garage space to it.
#2 Getting 4 people to commit the necessary money and time to it
If you want to focus on being on track but goofing around, don't go get a BMW and run Lemons with it. Go get a real, random, malaise era shit can and be silly and run Lemons and have fun. Trying to win Lemons races on laps is a waste of time.
If you want to go racing on a budget, true track time for dollar bang for the back to the max, focus on Chump car. Look at the car values given in their rules. Pick something you want to run, you think can be fast, and won't get you penalty laps from the onset, unless it's an E36, that can win with its penalty laps. Keep in mind when you look at this that you really want a manual trans car and that your fuel tank size is limited by the rules so a 4th gen F body with a V8 may be legal but a fuel stop is a mandatory 5 minute stop and it isn't going 2 hours (maximum driver stint time) between stops. You can run something like that and have fun running fast laps but just don't count on contending for the win. Also keep in mind that lighter cars with smaller tires will be way cheaper to run. Less fuel, cheaper tires, less brake wear.
The Lemons route you need to stick with a car that really passes for a $500 car or is just stupid and slow and will go through based on the fact they know it sucks. Count on having like $3000 minimum in a caged, safe, ready to race car.
The Chump Car route, don't get a rusty $500 car. Get a decently clean bodied car that costs a little more. Add money to the amount in the car to get it ready based on what you spent getting a little cleaner car.