What lens (if any) and external flash should I consider when shooting a broadway show?

I have a friend who wants me to do a shoot of a Broadway show she is in.

I normally do kids/family portraits and mainly outdoors, so I've never purchased an external flash.

I currently use a Nikon D40 with a 55/200mm lens. Will that lens work? If not, what type would I need to rent? Also, what type of external flash should I rent?

I really doubt you would be allowed to use a flash in any stage production. Perhaps not even a camera.
When I shoot events on a stage, I use a high ISO and fast lens with no flash.

If you DO manage to get your equipment into the show, the first time the flash goes off, you will be EJECTED from the venue.

Study and learn what you are doing before you attempt any important kind of shoot.