How to handle unvaccinated pets?

I require Rabies, DHPP, and Bordatella. I don't usually double book and have no dogs of my own, but I do work in a Spay/Neuter clinic so there's a small chance something might be passed. I also want to keep my environment as healthy as possible for the next boarder.

I require rabies vaccines be current for all dogs I watch, since that is a legal issue. I wouldn't be as concerned with the potential for disease transmission (not that it isn't a problem, but as you said, in most places the risk is very minor) as I would be with the possibility of being cited for possessing an unvaccinated animal in the event the dog did bite someone while in your care. Since the dog will be transitioning to a new person and a slightly different routine (and a new environment if you board), the dog will already be under more stress than usual, and a bite is more likely. Even if you feel prepared to handle a nippy dog, I'd still require proof of current vaccination. If an owner is unwilling to do that, it would tell me that they are not someone I want to do business with.

I don't require other vaccines, as my dogs are fully vaccinated, but I do inform owners that if their dogs haven't had their combo (DAPP/DAP/DHPP/ect) or bordatella that I will avoid locations where they will come into close contact with other dogs whose vaccine histories are unknown.

Same here. I require all necessary vaccinations and proof of it, not just word of mouth. I need to keep my own dogs safe and my environment safe as well. I was looking for a sitter for my own dogs when I went on vacation and I was surprised that I was not asked if my pets were vaccinated or proof of. Needless to say I did not leave my dogs there. The first thing I thought of was God knows how many dogs go through there without the proper health requirements. To answer your question I would not board the dog, in fact no one (professional facilities for boarding) will not accept a dog who are not vaccinated.