Hey, I applied to med schools last year, and around April, I realized I didn't have a lot of hope left in getting in anywhere 🙁 So I started applying for research positions around then. Well. I procrastinated and started sending out cover letters and CVs around May/June/July.
I don't know about everyone else, but I kinda did mine in a rolling fashion. None of the labs actually had a "job posting," I just emailed them and asked if they were willing to hire a "lab tech." Most of them weren't looking for anyone at the time (thanks to the terrible economy and the tight grant funding...), so every time I heard back from the "no"'s (or sometimes, you don't hear back), I sent out some more CVs. I think I sent out a whole lot in early May (for labs in my undergrad institution's med school), and heard back with ONE "maybe; we can solidify things in July ish." This worked for me because I was on a bunch of med school waitlists at the time and wanted to leave that possibility open. But this guy fell through later and said no. In late June, I sent out a bunch of more CVs, this time to HHMI labs (lab tech positions posted on the HHMI employment website), and after 2 interviews, I got a position 🙂 So I've been working in this lab since September, and it's been a really great experience!!
If you want a number, I think I probably emailed ~20 individual labs + 3 HHMI positions I applied for. (and for me, I'd say only about 10 individual labs got back to me, all with a no, except for the 1 maybe... for the HHMI positions, I got 2 no's and 1 yes)
But one thing to keep in mind... at least for me, I wanted a position where I could actually do experiments, not just be a plain old lab tech who's always doing dishes, making media or growing cells for everyone to use. But this also meant that the PI wanted me to commit for 2-3 years because getting me trained to do experiments would take time, and having me there for only a year would be a waste of their money and time (especially if I was gonna spend that one year busy with applications/interviews).
PM me if you have more specific questions, but otherwise, good luck!