The response from IAMSHANMAN is mostly correct. While official sources may tell you that you technically need a "work permit" to work, even if all of your work is remote, outside of Belize, the real story is that there are plenty of expats working remote jobs here while staying in-country on their tourist visa.
Once you have lived in Belize for 50 of 52 weeks you can apply for permanent residency. This process can take a year or longer. Once you are resident, you will get a Belize social security card that will allow you to be employed within Belize at a Belizean company, or run your own business doing business in Belize.
Shane Kenny International Living's Correspondent in Belize