If you're talking about animals and not all forms of traps employed against humans, you already answered your question with the most common word:
trapper
although @Traktor53 pointed out that specialists will call themselves X-catchers. (It's got a more positive spin on it than its synonyms X-trapper and X-hunter, since an X-hunter may have gone their entire life without, y'know, actually catching an X at all and an X-trapper's prey may have always escaped after being trapped.) If you want to coin your own word but want it to have some connection to English etymology, there's
trapsmith (who would build traps to sell to other people but not use them herself),
engineer (from an obsolete sense of engine as "An instance of artful trickery"), which could be distinguished by the obsolete or Scottish spelling ingineer,
awaiter (from await, obs., "An ambush, a trap, a plot"),
fanger (from fang, arch., "To grasp, to catch; to trap"),
stamper or stalper (from stamp, dial., "A trap"),
illaqueator (from Latin illaqueare, "To ensnare"), esp. with rope traps,
swarler or swarrer (from swarl, "To trap"),
swiker (from swike, obs., "A trick; a trap"),
braker, briker, or bryker (from brake, obs., "A trap"), and
panter or panther (orig. from Latin panthera, "A net for catching birds", but with a wider sense in English inclusive of treasonous plots).
There are some others even more misleading than engineer and braker, like lacer, latcher, leasher, puppy-snatcher, snarler, wirer, based on obsolete senses of those words. Again, you could create specialist terms like hutcher (an employer of hutches or "box-traps"), pitcher and picher (a setter of pitches or "fishtraps"), gnarer (a setter of choking traps), shraper (a setter of food-lure traps), and grinner or graner (an employer of grins or "rope-and-noose foot-traps"). The last could also be known as a swickler, after the noose itself. 'Kill-'em-all-and-let-God-sort-'em-out' is an exterminator.
As far as someone able to set mantraps, well, there is man-catcher (further buttressing Traktor's point) but for the most part people are a complicated lot and there are hordes of different skill-sets involving everything from rangers and commandos acting as ambushers to demolition experts variously blowing things up or stopping them from doing so to seductresses playing the part of a honeypot.
Side point: squire-trap for a bog or other soft piece of ground to catch the English landed gentry as they attempt to bother foxes is delightful and needs more usage.