I swear I posted a similar post a year or so ago, but I can't find it now, so I'm starting a new one.

I'd like to get my wife a bullpup style air rifle for chipmunks and squirrels.

I'm assuming at least that the bullpup-style rifles are more ergonomic and would at least feel lighter if not actually be lighter than a traditional rifle.  All the normal air rifles I've seen have such long barrels that there is no way she could lift and hold them steady long enough to aim it and fire.

Besides the fact that she is barely over 100 pounds, she has an old shoulder injury (rotator cuff) that she has never quite recovered from, so I need something easy to wield.  I believe the pistols would not be accurate enough for her and she would lose interest, so I'm hoping to get her a light-weight rifle that will be accurate enough to give her enough success to continue to use it.

I'm not looking to go overboard on this, but am willing to spend $600-$800 for a good system (rifle, scope, ammo, and propellant).

Idk. Lots of stuff missing from your query

Marauder pistol with a stock.
Hatsan flash pup

The Marauder pistol (P-rod) would be a good choice, and is fairly easy to fill with a hand pump.  It is not a real bull-pup but it is short and light with the stock, and it is rather affordable.

Although not a bull pup, I'd be looking at an Air Arms S200 for small, light, accurate, and high quality.

FX Wildcat PCPs are bullpup style and very light, especially the .177 or .22. Most of the weight is behind the grip so its easily managed by someone with little upper body strength.

Little over your budget but you can shop around and possibly by used. Air refills will be the tricky part. You can get 4500psi compressors for about $350.00 now or a used scuba tank if you have a place local that's willing to fill one for air gun use.

Check out the Ataman BP17

about $600 and 24 inches long -circa 6 lbs

available in up to .25 caliber

these are also supposed to be rebranded chinese made but with a regulator so you get more and consistent shot to shot power

PS - also invest in a sling and proper technique using it will go far to help her steady it

However, you'd need to pump it up for her or get her a compressor so that figures into the cost of a PCP air rifle