Construct a box using wire grates from a BBQ or similar heavy-duty wire grate. The box should have four sides and a back piece. Hold the grates together with zip ties, wire or by welding them. The box should be large enough and secure enough to easily hold a 10-pound turtle. If you know your turtle is larger, you may have to adjust for its size. Snapping turtles can get very large.
Construct a front gate with another piece of wire mesh that is slightly smaller than the opening. Attach the gate to the top of the box and push the bottom in several inches to create an opening. The gate should only open in, not out.
Hang a piece of bait in the trap. Oddly many snappers enjoy raw chicken and it has a strong smell so it makes a good choice. You can also use fish. Place your trap in an area where you've seen the turtle. You may wish to secure the trap to a downed tree with an extra piece of string or rope to keep it from floating away.
Secure empty soda bottles or milk jugs with string to keep the top third of the trap above water; one or two bottles at each corner should suffice. Turtles need air to breath so you don't want to trap them underwater.
Check your trap regularly. Once you've caught your snapper, handle it very carefully. Avoid the mouth and tail. Relocate your turtle to another marshy area. Use a stick to push the gate in and tip the turtle out.