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 would first send her an email asking when she might next be around campus and able to meet. If she responds that she's traveling (which is entirely possible now that the academic year is over), I think you can send a polite follow up asking her to be your official advisor.
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.