Hi Rick, welcome to the forum.
There are several companies nationally known that rent RVs (Cruise America is one I can think of) and there may well be some in your local area as well. I would caution you that the distance you're talking about means you'll quite likely be driving for close to two of those three weeks, noting that RV travel tends to be slower than car travel. And for that distance and time it can be rather expensive, too.
So a bit more information about what you intend to get out of this trip, whether it's visiting relatives, seeing something(s) specific at your destination(s) or perhaps general sightseeing along the way. You might also indicate any camping (tent or otherwise) and/or RV experience you have.
The most common "small RVs" from rental companies are pretty basic, since they expect a lot of folks with no idea of RV care to be renting, so don't expect luxury. But it can certainly be a fun trip.
BTW, I notice your "Location" is North Carolina, but you say "a trek from California to North Carolina" so perhaps this is a one way trip returning home from California?
In any case welcome, and hopefully we can get your question(s) more fully answered.