Welcome to the forums and please feel free to ask away to your heart's content.
There are more wildlife refuges than you might expect within a close distance to San José.
It seems that you have kids who need a good school? You might consider the British School in Rohrmoser or the European School in Heredia that both offer I.B. programs. (the International Baccalaureate is accepted widely in universities outside the U.S. )
So I don't see any problem with living in San José and volunteering with wildlife outside the city on a daily or part time basis. For example, the
Costa Rican Real Estate Group had listed the Genesis II preserve, very near the Tapantí National Park. There people would volunteer to help make or mark trails among other projects. Also, in La Garita (near Atenas) is Zoo Ave which is a rescue center and private zoo and I am sure would accept volunteers for a variety of needs. Carara National Park is also near San José.
I think there is a big difference between volunteer programs that are oriented for tourists. (for example the CCC has programs in Tortuguero where volunteers help turtles nest and count turtles - but they pay $75 a night to "volunteer" ) I think you can easily find on the Internet these tourist-oriented programs. As opposed to actual opportunities for volunteering, which aren't tourist oriented and are unpaid but also un-charged! These opportunities you can easily find once you are here. IMHO.
Your American-European broker will of course be a good source, but also you can find volunteer opportunities through guide books and also local media like the Tico Times. Also word of mouth will help, as you integrate yourself to the expat community you can learn about places to visit and eventually volunteer.