Do US military vehicles commonly have air conditioners?

No. Only vehicles for specialised applications would have them. Front Line Ambulances would have an air conditioned litter compartment, and vehicles carrying sensitive electronics would have it. The first "common" vehicle in the US inventory to come with it standard were the "uparmoured" versions of the HMMWV (the M1113 and M1114). Once the war in Iraq kicked off, they became more commonplace.