Perfectly legal to trap Mink, you can get a live capture trap at any half decent country supplies place, i have about a dozen traps set up at the lakes at any one time, best bait ive found to go i there is a tin of the cheapest tuna you can find, but only pierce holes in the lid, that way the smell escapes and attracts the Mink in and you dont have to keep rebaiting.
You will also you need to de scent the trap to get rid of any human scents, best way, is tie a rope to the trap and throw it over board for a week, this gets rid of any scents on it from manufacture and handling, then when you are ready to set it, wear a pair of latex gloves so you dont contaminate it, then when you have caught the thing, you need to dispose of it, i find a .22 pellet at close range works really well - dont throw the pelt away though, keep it and make something from it, i know you said you dont have a gun, but do you know anyone who has who could do the deed for you?