I don't know of any web based email apps that will let you connect multiple IMAP accounts, but i'm sure they exist somewhere.
Gmail has an option to continually download the POP3 mails from another account, but not IMAP. Gmail will allow you to import all mail from IMAP accounts, but it won't continually fetch new emails (except...there is an option to fetch NEW messages for 30 days...which you could just redo every thirty days)
Windows live hotmail does pretty much the same thing. You have to connect other accounts via pop3 ( i don't remember IMAP being an option) I haven't used Windows live hotmail in a while so they might have changed things.
For the server that your primary account is on, you should be able to easily enable pop3 (maybe configure gmail for pop3, but keep the downloaded messages on the server). What type of server is it?