I don’t want to seem as though I’m being petty by reporting her to whoever, but it is making my life a bit harder. I can’t tell her/trust her with the info. She’d tell the entire office and the depts outside ours and also probably tell everyone I don’t really need it, it’s just her personality
I don't want to be harsh, but I think you need to stand up for yourself here. There is a disabled parking bay and a blue badge which would help facilitate you being less disadvantaged by your disability, and at the moment you can't access it. If it was being used by another legitimate blue badge holder then fair enough to request another disabled space be created, but it's not. She's currently getting away with her illegal use of it because nobody cares enough to complain. You have a legitimate and good reason to complain.
Also, I think you're in a strong position because you already don't really speak. You've no need to worry if she dislikes you if you say anything, she's already not speaking to you! And who cares if she bitches about you behind your back? You haven't done anything wrong. Everybody already knows she's using a pass illegally!
I'd see if my line manager would have a quiet word with hers, and her line manager say something like "it's come to my attention that another member of staff has been diagnosed with a disability and it's triggered a review of our disability procedures. I'm aware you park in a disabled bay and I want you to know that next week going forwards we will be reviewing our disability policies and procedures including our car parking provision and inspecting blue badges to check they are being used appropriately. (If she knows manager knows) If it's still the case that you're not the legitimate holder of the pass you use, now would be a good time to stop using that bay before there are any repercussions, (or, if not known to be known by line manager) I don't have your disability on our records which you may prefer that way, but do let me know if there is anything that we can look at as part of this review that would support your work in this company.
Alternatively, just put on your brave pants and make it open knowledge that you have a disability. I have one, it doesn't make me a blue badge holder but it does affect how I do my work, and I prefer my colleagues to know so they don't think I'm getting special treatment or any resentment when I do things a bit differently. If you decide to make it fairly open policy, (you don't need to discuss more details than you decide to), it would take away any power she would have to gossip about you when you tell her you have a disability and need to use the disabled space. And anybody who bad mouths somebody with a disability is not going to come across well, either to colleagues or in the eyes of the company.
My take on it would be fairly brazen and matter of fact. You're legitimate in your need. She is not. She's going to see you parking in that space, she's going to realise it's you behind it if anybody tells her to bugger off using the space illegally, so I'd have a quiet word or email her directly (again, you only need to share basic information. "I am now personally a blue badge holder and need to use the disabled bay. I understand you borrow somebody else's badge to park there at the moment. If that's the case I'd really appreciate it if you would park elsewhere from tomorrow. Thanks for understanding") if it were me. It gives her the chance to rectify her parking without consequence. If that fails, then get somebody higher up to check her badge or report it to authorities.