We just went through the process in August of getting a permit and purchasing a cart. In order to submit an application for the permit, you must first obtain a Belize drivers license. This only requires copies of your passport and current US drivers license, obtaining a physical from a doctor in town, and, of course, paying the appropriate fees to the town council and doctor. Once you have your drivers license, you can apply for the permit which requires a copy of your new drivers license, completed application to include a picture of where you will be parking the cart, and again, more fees. The application will then go to the appropriate committee at the town council which will approve or deny the permit. If the permit is approved, it is valid for 3 months (make sure you have a cart ordered) and the cart must be inspected prior to being registered...again more fees.
I think have included all of the steps. Also, we are here under the QRP residency so not sure if there are any other steps involved for non QRP folks. Hope this helps.
Life is too short to drink cheap wine!