This is related to something I answered earlier today:
Mails are not receiving in Squirrelmail?
Except you're using Roundcube instead of Squirrelmail.
Also, take a look at:
as an overview of the various components of the mail system (i.e., MTA, MUA, MDA).
Roundcube, if I recall, is purely an IMAP client, maybe some POP3, I don't recall. In either case, you will need an IMAP server acting as the MDA.
That said, I can't think of any web-based mail client that will access the legacy mail spools off hand. It would be possible to write one, but that feels somewhat inflexible, given that we've already invented IMAP, which takes care of issues like simultaneous access on the spool file, etc. If you're running a for-real mail system in this day and age, you will want to use an MDA supporting something like IMAP.
There are also big, "groupware" systems like Horde or Zimbra. These will give you the whole mail system (plus other groupware applications) in one big package, so, in a sense, you'd only be installing one piece of software. But, under the hood, the internals will have the mail architecture concept I mentioned above, with an MTA -> MDA -> MUA.