Hi Hannah,
Assuming you are following the two week itinerary per your last question, leaving your luggage in Manuel Antonio wouldn’t make sense since you would have to go back to pick it up. If you are changing the order and flying from and then back to San Jose, you might want to ask your hotel there if they offer storage. Many hotels have a bodega or large closet that they keep things like that in. Just use your best judgment as for what you leave behind and obviously take your most valuable possessions with you. The other option is to just pay the extra baggage fee and bring everything with you.
As for an overnight stay in Corcovado National Park, you should definitely make your reservations ahead of time. A couple of great guides we know are Vincent from Etnias Tours and Javier from Discovering Corcovado with Javier. There are also many other great guides in Drake Bay but these are just two we know personally. For camping we’d also recommend you book in advance because they might need to plan for supplies if they are offering meals etc. Hope this helps!