The trap and/or shoot method has always worked well for us.
Tried relocating twice. Not going to bother again. The first time the same critter came back - they can travel a surprisingly long distance to find their home. The second time I trapped a groundhog for a guy I've done work for and relocated the groundhog 20+ miles away onto the back portion of our property. Wasn't long before we started having a groundhog problem.
But that got sorted out with a .22 pretty quick.
Used to have a raccoon problem too. They'd go down the street tipping over trash cans on a regular basis. Started setting a trap and whacking 'em first thing in the morn. IIRC, ended up whacking 16 of them in one summer. Neighborhood quieted down for several years after that.
FWIW, if you do shoot 'em, use a hollow point .22 or an "adult" air rifle (the ones that shoot 1,000 fps or faster). Tried using target .22 rounds and had a problem with shooting them two or three times and they'd still be alive. A good hollow point does the job in one.
If you live trap them, keep a barrel full of water on hand to drown them, or use a .22 / air rifle to dispatch 'em. No sense making it anyone else's problem. Oh, yea, and a large truck rumbling past does a nice job of covering up the sound of a small caliber going off. Pointblank you can use a target round or a .22 short or long (as opposed to long rifle) and just pop 'em once in the head.
I know it may sound cruel to some, but I had some groundhogs and raccoons tear up $8,000 of material in one year.