Any suggestions for ground cover for full sun - perferable one that stays green without flowering and requires little water to thrive.
Anything will need watering to get established.
Do you want evergreen?
a "meadow" of various short wild grasses and wildflowers?
I think your best bet is to use the wonderful search feature at the Missouri Botanical Garden website.
Put in your zone. Put in Full sun. Put in ground cover and see what it brings up. You'll be able to review good information about each plant type, narrow your preferences, and then work toward a list that you can take to a local nursery for further exploration.
Choosing a specific ground cover without any design context feels a little weird for me on this forum. I haven't a clue what or where or how this query fits within a landscape. Maybe it would make more sense to pave the area or use astro turf? Just kidding, but working in the dark leads to strange suggestions.
Here is a link that might be useful: Search Missouri Botanical Garden
I second the Mobot recommendation. Sometimes a search will return very few results. If that happens, just tweak the "Requirements" and "Characteristics" sections, as well as anything you checked in the far-right column, and search again.
After the Mobot search, you might want to ask here or elsewhere about particular plants.
For instance, some groundcovers are more aggressive than others, and they can be aggressive in different ways. So it may matter whether you're trying to fill in a 30' slope (you only have to trim the outer perimeter -- some people would use vinca for this) or the edges of your foundation beds (you'd need to trim along the lawn edging and make sure the groundcover doesn't strangle, creep behind, or climb up the shrubs -- vinca would quickly overwhelm your foundation shrubs and be making its way across the lawn in its mission to Take Over the World).
Not to be a smarta$$, but the best ground cover for full sun is grass. Not if you are a lawn maniac that wants a pure pristine mono culture and bright green all summer. A mix of clover, several grasses, creeping charlie, etc and mow it high and not as often.