This is one of the most commonly asked questions on this board and on the Belize by Naturalight boards. You might try the search engine to find other responses.
But basically the answer is: Yes and no. Yes, you can always find something at some price. No, it may not be your first choice or even third or fourth choice. The top quality places and the best budget places go first.
It also depends on where you are. In Corozal, Cayo and PG you certainly can find a place to stay, and there usually are a large number of vacancies even in the high season.
In Placencia and on Ambergris and Caye Caulker and other cayes, things can get booked up. A few times last year Placencia was nearly 100% booked at least in the budget and moderate places.
There's usually a slow period in mid-January, but at the end of the month things start getting busy, and that continues through Feb and March.
This year, due to Hurricane Keith, it's a little harder to predict what will happen, whether people will postpone trips or whether they will shift from the cayes, which got the brunt of the storm, to the mainland where there was little or no damage.
But, the bottom line is that in most areas you'll be able to find okay accommodation in January without advance reservations.
--Lan Sluder
Belize First Magazine
http://www.turq.com/belizefirst/