If you are coming in dry season there are very few, if any, mosquitoes on the island. The jungle seems to have very few mosquitoes at any time.
Malaria is rare, and the strain found in this region is both mild and easily treated.
Dengue is a mosquito-borne disease - we don't see much of it at all and again, it's a rainy season thing.
Ambergris Caye is a flat, sandy island. It has some areas of thick bush but we don't refer to that as jungle. The jungle is on the mainland, a significant distance from the sea.
If you want to see a lovely ruin and a beautiful jungle-river habitat, the Al-Tun-Ha day trip is a good one. I love Lamanai as well, but it is a longer day and your kids may get restless. Al Tun Ha is impressive and if your tour combines it with a stop at Maruba for lunch the kids can jump in the pool or ride horses there.
Have never heard of yellow fever in this area.
Always a good idea to stay current with tetanus - for daily life anywhere.
I've traveled to places where one is advised to have the full battery of shots and pills and stuff - I even got rabies vaccine once - but here I don't even bother with a flu shot. I think you will find yourself pleased with the healthy environment.
The Phoenix is superb - have a wonderful stay.