You can absolutely try to race, but based on the information you've provided, it is going to be a very rough go at it. As for the older equipment, you'll get a few odd looks but for the most part no one will care.
If you want to test yourself prior to the race, find the fastest group ride out there; I suspect the one you did probably isn't that fast (the tandems give it away). You'll definitely also want to get acclimated to riding in close quarters; for your own safety. Bike racing isn't like other sports, even the entry level of the sport is stupidly fast compared to the average recreational cyclist.