Just an advance warning, WIFI suks down there.. On some days they'll tell you it's not working or VERY SLOW because it's "cloudy" or they lost "satellite reception"..
There are plenty of hotels that offer free internet (not sure whether they're WIFI or plug-in) Some hotels, you have to pay around $5 dollars per day for WIFI. Or go to the nearest internet cafe (there are plenty of them in town) and pay to use WIFI.. There's one internet cafe in town that offers a free 15 min use of internet if you buy a drink.
I could only find one "budget" hotel that offered free internet.
Coconuts - http://ambergriscaye.com/coconuts/index.html