Idaho, Montana, and Wyoming. Idaho and Wyoming are reciprocity states, your state has to allow their residents to trap. Not sure about Montana, but I think it is also. Wyoming is reciprocal according to each species, for example Nevada allows nonresidents to trap most species but not bobcats or grey fox, so Nevada residents cannot hunt/trap grey fox or bobcat in Wyoming, but can other species. I believe that if a species isn't present in your home state this is not applicable, I know for instance that Florida residents can trap marten in Wyoming, because while Florida doesn't allow Wyoming residents to trap marten, that is because there are no marten in Florida, not because they are restricting nonresidents.
AK Tramp is correct, except Idaho just passed a regulation change (within the last couple weeks) so a client of an outfitter can dispatch a wolf in a trap (and I believe set the trap theirself) as long as they are accompanied by a licensed guide, without having passed the trapping course or the wolf trapping course. If you want to trap on your own or without a licensed outfitter you still must have taken a Idaho trapping class and a separate wolf trapping class (often offered back to back on consecutive days, or I've heard sometimes even offered the same day, now) those must be taken in person. That makes it a headache for someone from a long ways off, because they must sign up for a class (usually offered a few times a year), and travel to Idaho to take the class, usually not during trapping season, and then travel back again later during season to trap. We did have long seasons, September 10-March 31 on public ground in most of the state (select units had both longer and shorter trapping seasons) and year round on private, until last spring when the Grizzly lawsuit shortened them to December 1-February 28 in 3/4 of the state. Hunting seasons are still year round in the majority of the state.
Montana has an even shorter trapping season, Wyoming however has them classified as a predator (same as a coyote) in the majority of the state, except around the "Greater Yellowstone" area. Where classified as a predator it is open year round, no tag necessary, and no requirement to bring it in to have it checked and sealed, or even a requirement to retrieve it. You can legally shoot them and leave them lay, you are required to have a trapping license to trap, however.