Maybe a couple of times a week? It actually varies quite a bit. I always email her for schedule changes and I email her an agenda before each session. But I don't really count those.
When I am having an especially difficult time, like I have been recently it's more frequently. But I went through a long stable time where it was almost never between sessions. It almost always makes me feel anxious when I feel like I do it too much. She has been reassuring that I haven't been. But there is still the fear that this particular email is pushing it over the threshold. I do expect her to respond in some way. I don't expect her to do therapy over email other than maybe a quick reassuring statement. I haven't been able to pin her down on how much is too much, but she has told me that if it gets to be too much she will let me know and won't just stop responding.
I say if it makes you anxious at all you might as well bring it up in session. Otherwise from my experience it just eats at you