You
MAY be able to tow a trailer into Yellowstone before the first of July, depending on the route you take into the park, and the weather conditions between now and then.
According to skykingpilot "I say GO FOR IT! Who cares about the weather!" But you may have to do as I'm guessing he'd want to do --Take a helicopter
--if you want to be certain about getting into, and through, Yellowstone before July because it can be rather hard to negotiate those mountain passes if the roads are closed.
John
ps - I spend the entire summer and fall on the Yellowstone River about 35 miles north of the Park.