Will a Buick rendezvous run without thermostat?

It can, but you only want to do it as an absolute last resort, and only temporarily. On older, mechanically operated cars, the only consequence was that the heater wouldn't work. On newer, electronically controlled cars, the engine temp sensor will sense that the engine is running too cool, and will make the fuel/air mixture more rich. That'll reduce your fuel mileage, cause carbon buildup, and it'll plug up your catalytic converter in time.