Hey AirWis,
All nonresident big game hunting licenses in Wyoming with the exception of black bear and mountain lion are sold via computer lottery draw. The deadlines for applying are early. Elk - Jan. 31 (drawing held Feb. 28), Deer and Antelope - March 15 (drawing held June 30), Sheep, Moose and Mtn. Goat - Feb. 28 (drawing held May 1).
There is a Leftover License Draw held in July for Elk, Deer and Antelope but the leftover licenses are usually for private land areas where access is somewhat difficult. There are a few of these areas that will have licenses left over after the July drawing. It is rare that any of the good public hunting areas have licenses available after the first draw.
You have to be a Supreme Court Judge to understand the Wyoming Hunting Regulations. Best thing to do is get in touch with an outfitter or contact the Game and Fish Dept. in Cheyenne.
Good Luck!