I haven’t seen any camping in Ambergris Caye and I don’t think a lot of people do camp because it is a tropical island. Right now are the hotter months of the year so you would not be comfortable in a tent right now.
You don’t really go hiking or camping on Ambergris Caye; you have to go to the mainland of Belize to do that. Fortunately, Belize is a small country, so it doesn't take much time to get from one place to another.
There are no hills since Ambergris Caye is a flat island, so there is really no hiking here. You can, however, go to San Ignacio (the main town in the Cayo District of Belize), where there are a lot of places where you could go hiking.
You can also go to hiking areas in the southern part of Belize. You can go to the Mayan ruins where you can camp out. If you want to take public transportation, going to San Ignacio is a two and a half hour bus ride from Belize City. From Ambergris Caye, you could first take a 15-minute flight on a puddle jumper from Ambergris Caye to Belize City or you could take a water taxi, which could take an hour and a half to two hours. Once you are in Belize City, you could take a bus to San Ignacio, or you could get a taxi or rent a car. There are a lot of different travel options. Taking the bus would be the cheapest option but it takes longer.
(Pictured: Mayan ruin in Belize mainland.)