It (edit: my system) wouldn't do it that with a single login probably, unless it was a stored password on the PC for each mutt instance and data directory doing its connection to the provider.
Otherwise (with multi-logins), I could do that with my provider's email web UI in browser tabs (or save the logins in the browser, which I generally do not), for all the ones I designate to them as separate mailboxes (they have some number included, then you pay more after that), or could have a mutt script+window to switch to a different mutt data directory for each.
But I have the other backup stuff largely automated now, so I don't mind much. (Plus it makes me feel good to run scripts I wrote & tested carefully, on my PC.) There would certainly be simpler ways. Maybe I'll reconsider someday if I want to change. :) (Edit: for my own data, I avoid SaaS generally, as I like a sense of control. Same reason I guess that I go to the trouble to learn/use OpenBSD.)