I don't have experience in this since my T doesn't have an email where I could write.

However, I feel that you might get more out of this asking a different question. It doesn't really matter too much how this influences your T. Since it's her job, if it bothers her she has to change something about it, not you. However, it might influence you a whole lot?

I remember about a year ago I had the possibility to write to one of my friends as much as I wanted to, whenever I wanted to. He'd rarely respond, but that was fine. However, I actually found that over time, the relieve I got from messaging was less and less, while I was more and more preoccupied with the things I wrote. All my thoughts would only be focused on what I wrote and what I could write and how it made me feel. But it didn't provide actual relieve and didn't help me either.

If you are not dicussing what you are emailing about, maybe they might not help a whole lot? I think an important part of therapy is to learn to sit with your emotions and to wait for a week until you can discuss whatever needs to be discussed. Of course that doesn't mean that you always have to sit with your feelings. If you're having a bad week, you should be able to talk about them earlier. But also there, I'd feel that the issues should actually be discussed. Not just writen down without any kind of resoultion. Otherwise you might as well write a diary.

If it's something that bothers you, it might also be a good idea to talk to your T about it. Maybe she has some good ideas of how to start coping with less emails, and you guys can set up good boundaries that work for both of you.