Red, swollen labia, with no pain, and a virgin. What could cause it?

Seek evaluation. Without knowing more, this is difficult to assess. By itself it may not be anything, it could however be an early yeast infection (typically an odor, drainage and itchy) or an obstructed Bartholin's cyst (unilateral), vaginosis (discomfort, redness), simple overgrowth, hernia, etc. Treatment will depend on identify g the cause.
Infection/trauma... Infection of the area or trauma could cause what you describe. It should hurt at least a bit though. It could also be an allergic reaction to something.

Related Questions

Red swollen labia irritated vagina sensitive to touch white discharging forming around clitorus and labia no odor had unprotected sex a week ago?

Get examined. There are too many variables to give you an accurate answer without a physical GYN examination. Get one soon to determine the issues and obtain STD exam as well.