The reef is between 1/4 and 1/2 mile offshore. That is where the good visuals are.
Swimming to the reef will put you in boating lanes and that is deadly.
If you want to do your own thing, recommend that you stay someplace that is close to reef and has kayaks for guest use. You can paddle to the reef, tie off at a bouy and snorkel.
In the interest of safety tho, it is always recommended that you have a buddy with you.
Places that have kayaks and nearby good snorkel spots - El Pescador, Belizean Shores, Journey's End, Calm Caye Villa and Villa Del Sol.