Hi Hannah, 
Camping in June might not be too bad as long as you are prepared for some heavy off and on rains.

In La Fortuna you might want to check out Arenal Backpackers Resort. They have covered tents, each with a double mattress, outlets, etc. 

In Manuel Antonio, the only camping on the beach we can think of is Beach Packers, which is kind of run down. Also, I would worry about keeping your stuff safe since it is pretty wide open. But it is cheap. An alternative option for a simple hotel right on the beach in Manuel Antonio is Hotel Verde Mar.

Drake Bay has the Corcovado Tent Camp, which is luxury camping and very expensive but there is also Estacion Biologica Tamandua which is inexpensive, in a rainforest reserve, but not near the beach. You could also consider staying at one of the inexpensive options near the beach in Drake Bay and then doing one or two nights at the Sirena Ranger Station which has camping or bunks. Read more about that in our post Drake Bay: Costa Rica Unplugged.

Hope this helps with your planning. Have a great trip next week!