Is there a market in Richmond, VA that sells fresh lychee nuts (fruits)?

You should check the local Asian grocery stores. They are not always in season though, so not always found year-round.

heywinks makes a good point. The best Asian market that I know of is Tan A at the corner of Broad and Horsepen. If you don't find fresh lychee I'm pretty sure they'll have cans. There are also a couple of other Asian grocery stores in that area down Horsepen and a bit to the west. I can't recall the names though. Another place to try is Fresh Market which is on Parham north of Quioccossin. They have a good variety of produce and typically have things other the major chains don't carry. Along those lines also try Ellwood Thompson at Ellwood where it ends at Thompson (should be easy to remember). It's an organic store with a nice variety of produce as well.