Staying up all night, whacky eating schedules, running at odd times of the day, repeating it again-and-again, can make for a pretty tough experience. There are some things you can do (for a price) to make a relay race as "luxurious as possible." If you can afford, it why not?
Book one or more vacation homes to stay at before, during, and after the race. It takes a lot of planning depending on the course route but the ability to have a warm shower and a quiet room to sleep in is worth the price. Check out VRBO and other options to find the best places to stay.
Pro cycling teams bring along team chefs for big races. Why not do the same? If you don't have room for a chef because a motor home isn't allowed by the rules go for the next best thing. Map out a dining schedule of the best restaurants along the route. This isn't the time to eat sparingly. Relays are fun so eat dessert first, and last. This may not be ideal for your best performance but it sure is fun.
Most people borrow or bring whatever vehicle they have access to that can hold six people. If possible rent the biggest van possible. Twelve and fifteen passenger rigs aren't too big and they provide room for each person to stretch out during the race. If you want to be the envy of all the teams supersize the van and go for the customized Mercedes Benz Sprinter.