There are primitive camping spots all along service road 700. There is an area on the right just past the entrance to the main campground that is among some rocks which is not bad, except for noise from I-80. But it is closest to the main area.
There are others farther along the road, too.
They can be filled by people from Cheyenne and Laramie on the weekend, so I would get there early if you are arriving on Friday.
Since there will be no water, you can go to the visitor center at the Sherman Monument to get water, or use the restroom.
It's Wyoming and around 8800 feet, so the weather can be iffy at any time of the year. I graduated from UW and usually it's not bad this time of year.
WYDOT has webcams right across from Vedauwoo that you can check snowpack on if there is much left by then.
There will be mosquites and it will be soggy if you go climb on Valley Massif.
It's a mostly a trad area so don't go there expecting much sport. Beehive has moderate sport but the Zach O. guide is horrible for that area - I tried to make it easier to identify routes with posts on Mountain Project. (Which is interesting since he put in most of the routes.).