if it says vermin control for you 22lr then you can shoot foxes with it, what is short range and who polices this range.
Same person who polices every other range of shot. You!
I wouldn't want to shoot a fox even in a trap with an air rifle. very thick skulls in places and unless it is an FAC .22/25+ calibre of over 35ftlbs I just don't believe it is producing enough energy to give the fox an instant death without the risk of brain injury but not instantaneous death
I once shot a fox with a .22lr CBLong at about 20 feet
smack between the eyes
felled him on the spot with a very slight delay to him dropping
stone dead at the scene
on inspection the bullet had not penetrated the skull but internally had produced a massive haematoma
CB Longs produce 32ft/lbs at the muzzle (28ft/lbs at 50yds so no appreciable drop at 20ft)
a 30ft/lb FAC air rifle is not shooting solid lead 30gr bullet
but often readily deformed pellets of 15-20gr (in .22 form, less in .177)
as for licensing
no-one cares, (no legal precedence, no legal definition of fox as vermin or more to the point NOT as vermin, no-one public defender with the balls to test it without numerous other crimes to lump together)
I have fox on ALL rifles as a matter of course as only because I do it regularly in urban environments
prior to this I would always shoot a close fox with the .22 if out rabbitting