Hi,
I don't mean to minimise her problem, but both my DC went through exactly the same hermit stage at that age - the summer of turning 14. Saw no one all summer. A close friend said both her DTs did too, even one who is close to my son and they live round the corner from each other. I was worried but pretty reassured by my friend whose sons are older than my two. I ended up calling it their cocoon summer. It's as though they need a summer to themselves to make that transition.
Having said that, I did a few things to prevent the summer being a wash out. It helped in our case to be really direct with them. I said something like, I'm worried you've been isolated this summer and you seem a bit down. I don't want to intrude but it's my job as a mum to check you're OK, so I have to ask and really want an honest answer: Do you feel you've fallen out with friends or you're being excluded from stuff you wish you were invited to?
You can often tell from their eyes if they are lying. I once nudged DS2 when he said: no, I'm fine, and I said: well, you say that, but your face doesn't look fine. You look about to cry. And he burst into tears and told me he'd fallen out with friends. Then we talked it over and suggested some solutions and he felt better.
But if she says she's fine, she just doesn't have anyone she wants to hang out with (which is what both my DC said at that age) then imo, it's still fine for you to say, 'Well that's good to know, but it's not healthy to be in your room all day long' and discuss some stuff you could do together.
Also, are there no summer schools for dance or sports that she could sign up for?
I looked up summer bucket lists and music festivals and discussed skills they wish they had then drew up some easy things to do - we went to a music festival, to a gig, to a couple of workshops and masterclasses in stuff they love (music and sports) for a couple of long walks, to sightsee round a local city - just a few things together. Long term it doesn't matter much if they have a summer hanging out with their mum. The following summer I never saw them, they were off doing summer schools and out with mates and the social life had kicked in.
We also did a few things at home. It was a good summer to say, well, at some point in life you'll need to know how to load a washing machine/cook a few healthy dinners/iron a shirt properly so you may as well learn now. They're still just about young enough to tolerate that being a bit of a game.