I can't think of anywhere near by. The north platte is really good but it is about 1hr 30 from Laramie. That area is really beautiful but a tad hot in June. I don't fly fish but if any others come to mind I'll let you know. Might be some stuff on the eastern side of Medicine Bow. That'd be 45 mins from Vedauwoo.
Laramie River is about your closest option, but next month will be the peak runoff, so it might not be fishable. There are a few beaver ponds in the glen that might hold a few small brook trout, but I can't recall ever seeing a "head" rising in any of them. Vedauwoo is not the best place to play the "climb a pitch, catch a fish" game.