Hello friend ---- a visit to Belize is a wonderful thing, but your work exchange idea may be difficult to implement. You need a work permit to work here, even when it's work-exchange as you suggest.

There are some charities and NGO's in Belize that have excellent volunteer programs, and they make the appropriate arrangements for their volunteers. The details on those and how to participate would be found via the websites that promote general "vacation-volunteering". In most cases the volunteers pay to do the programs (not a lot but something) rather than receiving food & housing "in exchange". You might want to check out the websites that do such placements and see what is required, how to apply and whether food and housing is included. Most of these programs offer excellent experiences.