I don't see
why Flash would be involved in webmail.
What webmail account are you using? Have you tried going to your webmail login page on your own, (without waiting for an email notification), and logging in?
What is he exact wording of the error message?
Are you aware of scams? I don't know what it is that is telling you you have new emails, (my web mail doesn't do that), but I will always visit my web mail login page manually and then log in to my web mail and check mail that way.