OK here is the deal in Wyoming. All antelope licenses are by drawing. If your a non-resident you apply in Jan or Feb. About 10% of the total available licenses are awarded to those that are not residents of Wyoming. Some units have more licenses than applicants. Those are available for purchase on a first come first served basis after the drawing. This year about July 8. The resident draw is done after the non-resident draw.
Most areas that have licenses left after the draw are for units with limited access. Lots of private property. Getting harder to get access. Sometimes access is available by paying a "trespassing fee" which allows access to a private ranch. Sometimes the game and fish dept. has lists of farmers that will allow access. Often this fee is less than $200. Lots of the landowners, especially bigger properties, have leased to outfittors. The outfittors guide hunters for a fee on those ranches. They are normally the only ones allowed to hunt that property. Non-residents can hunt for free on public land that has public access. Please note it is illegal to cross private property to access public land without permission from the landowner. In my local area the antelope get hit hard on opening day and usually move to private property to avoid most of the hunters. Some of our units have enough antelope that you can buy 2 licenses for any (buck) antelope and 2 for doe/fawn. This allows a bit more hunting on a trip to Wy.
A bit to the SE of me near Newcastle there is a bit more public land with better access than in my immediate area. Not sure if there are any leftover licenses for that area, You could easily check on line. Be aware that some units have limited antelope habitat. They are a prairie animals and don't live in steep mtns and heavy timber. So you have to be sure there are animals where you intend to hunt.
We have lots of antelope in our area. Most of the land is private. I have access to a large ranch that has about 70-80,000 acres. The ranch manager has become a personal friend. He doesn't like antelope as they tear up fences. I would be interested in trading a hunt there if someone has one to offer. This ranch would have other hunters on it early in the season. Less later on. There are always antelope somewhere on the ranch. The ranch is half in 1 unit and half in another. IF we get a good snowstorm in Oct it will cause the antelope to herd up, usually on his ranch. Have seen several hundred in late Oct when we have snow.
In my part of Wy there are VERY few true trophy antelope. Lots of good mature animals that are representative. Other parts of the state grow some bigger ones. For a book buck look at New Mexico.
Deer are done differently just to make things more confusing. Some areas hold mostly mule deer and some mostly whitetails. Deciding where and how can be a bit of a challenge. Lots more public land where there are mule deer. They tend to like the mtns. Some hunt units are by draw only and some have no quota and are called general hunt units. Residents can buy a license for a general hunt area anytime. There is a limit of one per hunter per year as a general rule.
If there is a limited quota then you have to apply just like for antelope and the same rules apply for leftover licenses. For a non-resident to hunt a general hunt unit they have to apply for a license that is good for that particular region. Then they can hunt any general unit in that region if they draw. The F@G limits the number of licenses available to a non-resident. I have places to take someone in both the Bighorns and the Black Hills should a traded hunt be available. Otherwise I will give some suggestions in either place where you might find a deer. To hunt in Wyoming's Wilderness area's a guide is required. Any resident can apply to be a "resident guide" for a non-resident hunter. No $$ can change hands unless they are a licensed guide. Then those hunts can add up. My son and I signed up to help a friend with some elk hunters on a pack-in trip into the wilderness years ago. There is a lot of public land to hunt deer on in Wy. Hiring a guide or paying a trespass fee increases the odds of finding a animal. You can do it yourself with a little research. Best of luck. Bruce