My T allows me to email as much as I want (he even specified want as opposed to need) BUT in my situation I need to understand he may or may not get a chance to read them that day or even before our next session. He does promise he will read all of them eventually and that if he has not read something I need him to read for session I can tell him which one and he will read it in the beginning of session.
Since the pandemic I have really only had H to talk to so poor T has gotten a lot of emails. I did clarify with him what my expectations are when I send him an email (I don’t care if he waits until session to read them, I don’t expect him to read them during his free time and I will specify if I would prefer a reply). He has had no problems with all my needyness.
Other T’s have made big promises and then gotten angry though so...