You can come sorta close to what you describe, but not exactly. You can't access the Thunderbird database from a webmail service because it doesn't work in that direction. So if you download your messages with POP3, delete them from the provider server, and then do stuff to them in Thunderbird, the result isn't accessible from a web service.
You would need to keep your messages at each provider's server and do the work there through Thunderbird and a webmail service as "front-ends" using IMAP. Thunderbird can do that, and some webmail providers can also. I'm pretty sure Gmail is one that can act as an email client for other providers' emails.
So you would set up say Gmail similar to how you set up Thunderbird, using IMAP to interface with the other provider accounts. You would then have the equivalent of a single access point, usable from both Thunderbird and, in this example, Gmail.
Note that this applies only to new messages and old ones still on the providers' servers. If you wanted to include old messages that are now only in Thunderbird, you would need to look at export options (native or third party), which would depend on the webmail client. You may be able to export the historical messages and import them into the webmail client so they would also be accessible from there. If a compatible export option isn't available, you could do something like resend the old messages to yourself. Note, though, that this will create duplicate messages that would need to be managed in two locations.