Just a note of clarification for the regs that wytex posted. You must send a new notarized affidavit and set of current regs from your state every year with your application and if you state prevents Wyoming residents from trapping certain animals, you can still get a Wyoming license, you just are not allowed to harvest those specific species. On the other hand, from my understanding if your state allows Wyoming residents to trap whatever species are legal to trap in your state, but there are species in Wyoming that your state doesn't have, they will let you trap them in Wyoming. So if you are from, say Florida, which doesn't have marten, as long as Florida allows Wyoming residents to trap whatever furbearers they do have, Florida residents can trap marten in Wyoming. Also, to harvest furbearers by hunting in Wyoming you must have a Wyoming trapping license, and the same regs apply, so if your state won't allow Wyoming residents to trap that species, you can't hunt or trap it in Wyoming. The one example of this that comes to mind is Minnesota, which doesn't allow nonresident trappers, but the last I knew did allow nonresidents to hunt bobcats. But since they don't allow nonresident trapping Minnesota residents cannot hunt bobcats in Wyoming, even though a Wyoming resident could go to Minnesota and hunt them.
Also, give yourself plenty of time when sending in your application, sometimes Cheyenne takes their time processing applications and getting the new license in the mail. I've had it take 2-3 weeks before, so don't wait until the last minute.