help with my emails

[Cupcakemumma]

New to the board.
I have a website hosted by a provider and unfortunately they are not answering my emails.
So I'm thinking of changing my website provider but is there any way I can still access my round cube hosted emails.  I have the password but I'm trying to log in and I get this error.

Does anyone have any suggestions as to how I can access my emails or are they lost

Any help would be appreciated.
I am not completely computer illiterate but I'm not super savvy either. 
Thank you so much for any assistance .