My grandson's john deere 4x2 gator works fine in 1st & 2nd gear but will not reverse. It does not make any sounds other than normal, but gear won't stay in reverse and if you hold it down it still goes forward.

SOURCE: John Deere Gator 4x2 - Left Wheel doesn't move when in 2nd gear

I would replace the high/low switch and the reverse switch. One time a replaced the high/low switch and the reverse switch quit on me 2 weeks later.
You can also get the Peg Perego Gator Shifter that upgrades the vehicle.
You can get the parts and repair help at this site
http://www.shopjandl.com/kids---baby-ride-on-toys-peg-perego-switches-and-shifters.html
They have helped me several times.
Hope this helps