As already pointed out, IMAP was made for just this scenario, and should be fine.
Outlook can be used as an IMAP client, but Thunderbird would also be an excellent choice, possibly better, because Outlook is really meant to be used with an Exchange server.
Another possibility would be to use webmail, where everything is centrally stored. You could use any commercial provider (such as GMail), or if you prefer, set up your own server (or have it hosted). There are many excellent mail servers and webmail packages out there, many free software (such as Postfix, SquirrelMail, RoundCube).
The two options can also be combined (probably best): Most webmail systems can themselves use IMAP (sometimes it's even the default), so you can use IMAP and webmail alternatingly, for maximum flexibility.
As to outgoing mail, most IMAP clients allow you to automatically store a copy of outgoing mail in a folder on the server (either special "Sent" folder, or in the folder you currently are in). Then copies outgoing mail will be available just like all other mail via IMAP. I use this myself in Thunderbird.
In addition to mail, your client might make good use of some kind of groupware/collaboration software with shared calendar etc.
See e.g. http://serverfault.com/questions/5282/is-there-any-open-source-exchange-server for some options. Or maybe a wiki is more appropriate. At any rate, having more than just email for communication can be very beneficial. Which solution is appropriate will however depend on the exact kind of work (regular office work, technical support, sw development, creative work, research...).