Is there a way to get put on Active Orders in the Army Reserve for the summer? If so, how and who would I contact?

Just run it up with your chain of Command. They can go over your different options.

AT Orders. ADT Orders. ADOS orders. Title 10 orders. Yes, there is a way. What the reason is for will determine the type of orders and the pot of money the pay will come from. And you talk with your unit.