What's the best way to seal an access hole in a wooden gate?
If your goal is to make the hole look "neat" instead of just a chunk cut out of the fence gate, I would suggest simply adding a little trim around the hole like was done in the picture below.  Obviously your hole is probably smaller, and a thinner trim would need to be ripped from fence boards, but the concept is the same - cover the rough, unfinished edge of wood with nice, sharply cut trim.



Another option would be to grab some "outside corner molding" which is sold at most stores that stock interior moldings.  It would need to be treated to handle the weather.