"However, I have her Facebook and Instagram info, so I can message her
there, but is it appropriate to do so?"
Yes, it is "appropriate" to do so. In other words, you would increase the chances of getting a letter if you do so, and only very marginally decrease it. The reason is obvious: you make her understand that it is important to you.
Furthermore, Academia is not a usual work-place. It is certainly not a corporate work-place, and for most parts it's not even a "work": many of the major aspects of academic life are on a volunteer basis. Hence, while it may be seen as "unprofessional" to contact your boss on Facebook if you work in a traditional industry/corporate (e.g., insurance company), it is not entirely inappropriate to contact academic colleges on social networks.
Clarifications: it is impossible to know in advance if your supervisor is going to be happy or not to receive your message.
It is also important to understand different cultures, and know where your previous supervisor works: if she is from UK/US, then she may indeed be more inclined to see her job as akin to a "corporate" job, IMO. If she is from Central Europe, she may be less so (again, IMO).