Several questions:
1. How many email accounts do you have? Are you only dealing with one account or do you have several accounts that you want to view in this web-client?
2. What is your present provider? Gmail? Outlook.com? Another?
Observation:
This doesnt' answer your main question about an online webmail client (sans email account), but....
If ads are your main concern there are some ways to remove ads. If you're using Firefox/Palemoon there is a webmail ad
blocker add-on that removes the ads from Gmail, Outlook.com, etc. In fact I keep forgetting there are ads in these providers, since I've had that add-on for some time and never see them! Btw, it's also available for Chrome. You could try this and see if it makes your present webmail provider/experience more bearable.
If it does work for you and if you have more than one account (at other providers), you could set up your main account to POP your other accounts (or have the other accounts auto-forward to your main one) so that you can do all your reading/organizing at that one location, rather than relying on an general IMAP web-client. And you can set that main account up so it can (probably) send mail using any of the email addresses of your other accounts.
You could basically 'live' out of this main account and ignore the others, at least for a little while. You would have to login to those other accounts every once in a while, though, so they won't be suspended. And you wouldn't have the convenience of an IMAP connection to move messages around those other accounts as easily as with a true IMAP client. But you might be able to get by.
As far as an actual answer to your question, I'm afraid the one service I mentioned is the only one I ever heard of, barring those premium email providers (e.g. PolarsMail, EuMX) that use something like Group Office software, which can be set up with an IMAP connection to other, external accounts. (E.g. I have my EuMX Group Office webmail set up so it can work with Fastmail and Runbox via IMAP.) Accounts at PolarisMail and EuMX aren't free, but
very reasonable (less than the price of one home-delivered pizza covers a whole year of service, including good customer service).