Wyoming has way more goats than any other State. There is lots of Government land in the west side of the State, but licenses are hard to draw there. Some residents go several years between licenses in the more popular areas. The east half of the State, say from Casper east is largely private land.This is the situation like has already been described,with licenses left over after the drawing. If I wanted to be sure, I would put in for the drawing in some of the eastern areas. If you wait for leftover licenses there is a chance you may not get what you want.
A pre trip, even before you apply for the license is the best if you can swing it. Drive through some habitat and stop and talk to landowners. A lot of them have hunters that come back year after year, but some have room for a new outfit. I suppose most want a trespass fee, I don't know much about that, but I doubt it is very high. There are walk in areas where you don't have to have permission to hunt, and State land, but State land is almost always only 640 acres in any one parcel. If you have some map reading skills you can get a goat from them but it takes more effort.