The faculty, of course, will not first hear from me over email - we have maintained semi-regular research-related meetings over the semester. I have not yet asked her to be my advisor, but now since the academic year has ended, I want to make this an official advisor-advisee relationship. I'm wondering if it would be bad form to just ask the question directly over email, as she doesn't seem to be around the department anymore. I should add that I am already a PHD student in the department, and funding is provided by the department, so I am not really asking for money either.
I don't think it can be construed as bad form to contact a potential advisor by email. This is the way that I approached my advisor. I had completed a undergraduate course with him but he would not have remembered me due to class size and time lapsed. I drafted an email explaining the course I had taken under him, gave a very brief outline of my research topic (about 5 lines) and had another couple sentences as to why I thought he would be a good match as an advisor for that particular topic.
I would not expect (as in my case) a yes or no reply but if they are interested it would be assumed that they would email you back either looking for some more detail or to set up a meeting to discuss the proposal with you. You may have to wait some time for a reply but you could leave it a couple of weeks before following up. It may also be good to check with the department in case the person is on leave etc.
I don't think it's bad form at all - as long as you're concise and polite. I asked my PhD dissertation advisor via email (like you I already had a relationship with him) and he actually responded thanking me for asking via email, rather than springing it on him in person (where I guess it might be harder to say no or think about how it would fit into his current workload...). I just wrote a short email reminding him of my research topic, laying out why I thought he would be a good match, and asking him if he would consider being my advisor. He said yes via email, and so far (I'll be completing next year) it's all going great.