Aye
For close range work, A 22lr with high velocity ammo (laser etc) will do the job ,but you must be certain of your shot and always hit a kill zone (your neighbours will be less pleased to find a wounded fox lying about)
This will be a fair bit quieter than the 12 bore. The shotgun will also be up to it if you can call the buggers in close enough.
Have to say though, I use a 243 for foxes and it has to be a pretty close range shot for me to take it with the 22lr. I`ve met a few who go on about 100yrd+ foxes with the 22lr,"head shots only mate ,,,easy,,don`t need anything bigger" , but