You can live trap them but it requires a special trap and they run about $350. Snaring would be your best option if you don't want to wait on one to shoot it. You can get a dozen snares for about $15. There should be "slides" areas where the beaver are coming out. They will look like little mud slides, hence the name. Hang a snare over the slide with a 10 inch loop 3 inches off of the ground. A beaver has a larger neck than head so they need to get at least one front leg through to hold them. I prefer washer lock snares for beaver, they will relax and won't hurt the beaver. If you get a small one you can let it go unharmed and dogs will also not be killed with the washer locks if that is a concern.
If you don't see any slides, make your own with a shovel and place a few fresh peeled sticks behind the snare with a little beaver lure on them. Beaver are territorial and they will go right up to check these sets out.