That's odd - the last I knew Sumner sportsman's club didn't have any skeet houses ..... although they do have Trap and sporting clays once a month / and a make shift 5 stand that operates once a week or so.
You're not going to see any serious Skeet shooters hanging out at Sumner / they're mostly Trap shooters - so you won't see hardly any sub-gague guns there.
But picking a gun to fit her - is probably more about getting something that doesn't intimidate her than anything. But 5'9" and 125 lbs at least - means she can handle whatever she wants / but all the more reason to focus on a 20ga for her. I think a semi-auto will be perfect for her / or maybe a 20ga O/U with 28" barrels ( something around 7 1/2 lbs ). A Beretta, Benelli, Remington, etc depending on your budget will all work. My pesonal choice is a Benelli SuperSport ( Cabelas, Western Outdoors (used to be Sportsman's warehouse in Fife or Lacy, etc have them in stock for around $1,750 ). The 20ga version - is very light at about 6.2 lbs / and even my 13 year old granddaughters like shooting that gun. I also have a Browning Citori XS Skeet, 30" barrels, in 28ga that I use when the kids are starting out / but that's a $3,000 gun these days if you can find one.
I think focusing on a semi-auto is probably smart / and you'll find a lot of good used guns around too.