Sure there's a need for car seats for small children. There are more than a few bad drivers in Belize, and the incidence of driving after drinking is probably higher in Belize than in the U.S. Belize has a seatbelt law which requires seatbelt use (though it's not routinely enforced.)
I don't know how many of the car renters have child seats, but the "majors" such as National, Budget, etc. in Belize City should be able to provide them. I'd suggest contacting them in advance.
Belize First Magazine http://www.turq.com/belizefirst/