the hardcore way is to set up an smtp server on your 24/7 system and forward all mail from yahoo.com to your own machine.
the slightly easier approach is to use fetchmail to automatically poll those acounts every 10 minutes or whatever, and get new messages. this requires the webmail site to have either a pop3 or imap server running, which yahoo seems to: pop.mail.yahoo.com
this will put all your mail into the system mail folders /var/spool/mail/username which is what conventional programs use.
you can use mozilla or somethign like that... but it's not as fitting with the linux ethos i reckon.