CompleteProject will work with any e-mail program - including online services. However, if you want to use the convenience email buttons within CompleteProject - such as New and Reply - you will need to install a local e-mail client. Fortunately there are loads of great e-mail clients out there - some of which are completely free.
If you do not have a local e-mail client installed we highly recommend you download and install one. Note that local e-mail clients can be used to check online e-mail accounts and modern e-mail clients have very simple installation and configuration procedures. So having a rich local desktop e-mail client is a great way to improve your productivity - even if you are not a CompleteProject user.
We can recommend the following free e-mail clients:
Windows Live Mail from Microsoft is a great e-mail client with loads of features including dead simple installation and configuration and the ability to support multiple e-mail accounts.
Mozilla Thunderbird is a great open source product from the guys that brought you Firefox. Thunderbird is simple, easy to use and even comes with junk mail filters. This is probably the most popular free e-mail client.
If neither of those e-mail clients meet your needs check out the Opera Mail client - another solid, well tested and easy to use e-mail client. Alternatively if you are absolutely set on using a webmail client then be aware that solutions exist which allow you to use a webmail service with the mailto protocol. For example MailToUpdater can be used to set Gmail as your default mailto handler.