If you have the budget, there are certainly gentleman-racer seats available in most series. I don't know what the budget is (I know if I have to ask, I can't afford it) but my guess is that for a season its High 5-figures to low 6-figures for a season with a team in NARRA or equivalent.
If you're not interested as much in prize money and TV fame, troll around NASA, SCCA and other race-sponsoring but still-recreational organizations. There are teams...especially for Enduros......that welcome additional drivers (and their money).