I'm actually trying to figure out if ladies ever drove themselves places, as well. Like, a gig only held two people, from what I'm reading. I'm not certain whether that includes the driver, but if it doesn't, I suppose a lady just wouldn't ride in a gig? If anyone knows, please enlighten me.
It depends on how you decide to write your FMC. How outrageous do you want her to be? How much would her family let her get away with?
Women could drive their own carriages. BUT they'd have a groom there with them (as would a man). The groom's job is to drive when necessary, jump down and hold the horses when the driver wants to stop somewhere, and could count as a chaperone depending on her family. Many carriages had a small seat in the back for the groom to sit on. Or even just a shelf to stand on with a hand grip.
I doubt your fashionable lady would be driving a gig by herself (with groom) in London. I'd go with a phaeton or curricle. But she'd have to be a fairly good driver to handle even a pair of horses, especially in London traffic. Having her groom drive her with her maid next to her would probably be fine. Even better if it's her chaperone, mother, grandmother, aunt, or older married sister.
How wealthy is she or her family? An unmarried woman owning her own personal carriage would be quite wealthy. If she wants to promenade on Rotton Row in Hyde park, being driven there by a groom or driver in company with a suitable chaperone is completely unremarkable especially in the family's baroche or landau (check dates on when which kinds of carriages were popular and with whom). Walking in Hyde park (not on Rotton Row, but there are lots of foot paths) would also be unremarkable and more easy to pass a note.
She could also ride on Rotton Row if she's a decent rider--and riding side saddle requires IMO more skill than riding astride. With groom in tow.
If her family approves of the match, she mightn't need to pass notes. But if she does, other places to do it--provided the two of them could work out a time for both of them to be there--would be the Panorama, Somerset House (viewing art), a balloon launch, at the lending library, at a confectionery shop like Gunthers, The Royal Menagerie, The British Museum, if they're both at the same party or musical offering. Or if her maid will help her pass notes.
If her family is fairly conventional, she could no doubt go places with her maid and a groom and/or driver as chaperones. But most places she'd be with whatever older woman is in charge of her--mother, hired chaperone, aunt, etc.
Let me gift you with
The Regency Redingote. She's a bit repetitive and I could wish for more citations but what I've checked has been good data. And she delves into less well known areas of the Regency. Here are the articles in her
transportation tag.
Try looking through Wikipedia and Google for novels and diaries and books of letters written around the time you're writing about. That will also give you ideas on what would and wouldn't be acceptable. The Internet Archive and Project Gutenberg have good archives of books written during that time that are now out of copyright so available on line and for free.
Have fun!