I don't know whether you'll be able to find something for four people up in the mountains within your price range. If you end up having to expand your possibilities by looking down in the valley (not the usual recommendation, but lodging four people is pricey), I want to warn you that you may well need to have air conditioning. Unfortunately, a lot of the lodgings in the valley do not have a/c. They will tell you it's not terribly hot very often, but that is of no help to you if it is hot while you are there. Up in Ortisei (and I assume also Cortina), a/c would nearly always be unnecessary.
Bolzano, the largest valley city, has a very beautiful historic district as well as the museum housing the Iceman. However, I don't know that air-conditioned lodgings there would be any less expensive than non-air-conditioned lodgings up in Ortisei. Obviously, it's better to sleep up in the mountains, but there is decent bus service from Bolzano to Ortisei.
Oops--Overlooked the "with car" comment; I don't know anything about the parking situation in Bolzano, but it's not a high-rise city, so I doubt that it's too much of a problem. But with a car, I think you can find a place that's workable and closer to Ortisei.
