I'm currently thinking about applying for PhD scholarships, although I'm not really interesting in starting this September, so I'm mostly thinking about the future. I've found two academics whose research interests fit mine, both at the university closest to my home growing up (I know this shouldn't be an important consideration, but I would like to be close to my family while doing my PhD). Neither have current students listed on their staff pages, but neither do they say anything about potentially supervising students. Their areas of expertise overlap somewhat, as well.

Is it advisable to email them both asking about potential supervision? Would this make me appear less focused or interested in their research? Or would a general email to the head of postgraduate studies or whoever fills a similar position be a better idea?

I would email both and see if they would be willing to meet for a chat about your ideas. I personally would not have an issue with this if you emailed me and another colleague directly - but make sure they both know you emailed the other! Although if your looking for a funded PhD, they are not likely to be in a position to make these sort of offers. At my university (a post 92) this is a faculty decision with the studentship (no matter now funded) being externally advertised and recruited to by HR.

Is this in the humanities or social sciences? It would be pretty unusual not to have any students if it was the sciences, and it would probably mean they are not in a position to take on students.