I've stayed in Riverton for a week at a time for work and it seemed pretty laid back to me. My understanding is that Riverton has higher crime than usual.... for Wyoming. The area is absolutely gorgeous. I honestly don't know if Riverton is technically part of the Wind River Reservation or not. Awful lot of pale faces live there if that's true.
As far as carrying, I've found the following information:
http://qcode.us/codes/riverton/
9.20.020 Weapons—Carrying.
No person shall carry, within the city, a deadly or dangerous weapon concealed on or about his or her person unless they are a peace officer, a person holding a valid concealed weapons permit, or persons in compliance with Wyoming State Statute 6-8-104. Any person who violates this section shall be deemed guilty of a misdemeanor, punishable by a fine of up to seven hundred fifty dollars ($750.00), up to six months in jail or both. (Ord. 11-007 § 1, 2011; Ord. 02-006 § 8, 2002; Ord. 98-007 § 53, 1998; prior code § 16-108)
http://law.justia.com/codes/wyoming/2011/title6/chapter8/section6-8-104/
The Wyoming state statute that's referenced above sounds like the Constitutional carry provision to me. Bear in mind, I don't live there, I've never spoken to a LEO there and I am not a lawyer. Wyoming is a preemption state so local municipalities can't legally craft ordinances regarding possession, purchase or carrying of firearms. Again, not a lawyer.