I am new to the ford 9". I found a good deal on a slightly used ring and pinion. I was wondering if there were any problems running a lightweight r&p in this axle? Or if richmond was any good for a non drag vehicle.

Sure. Run it.

The Richmond r&p should hold up fine in the off road application. Points of interest, you did not mention what kind of rig this is going in? Be aware that even a richmond set up correctly with have some pinion whine-not a personal issue but some complain. So, on the street if it is going on the street much you will hear the whine, that depending on your exhaust system and right foot :) I ran one in a 408 stang back in the mid 90's and it lived for 2 seasons of street/strip abuse-sold the car after that.
HTH,
KBebsss
NC

Its in a jeep that is mostly off road, I drive it around sometimes, maybe 2k miles a year on road.