Is there any software which allows visitors to sign up for an email account on your website

Yes, there are many products outside. Guess you want to go with something free, try SquirrelMail (http://www.squirrelmail.org/).

However understand that implementing a webmail solution is always kind of risky, especially you have an open source product that comes without support or warranty.

You may want to try a larger hosting company that may have that included in their package deal. When you start giving out e-mail addresses they have to be setup on a mail server with SMTP running and lot of other neat things that go into sending and recieving email.

ispConfig has this feature but it's more of a developers type of program. It has all of the major applications necessary to run a hosting company including email, ftp, web, chat, etc.

Hi, Just so you know quite a few hosts have it in their terms that you cannot freely give out email addresses to the public for security reasons (spam etc) so you should really check out your hosts terms of service in detail to make sure your not breaking any rules, generally i would advise a vps/dedicated solution because then your are in control of the server and wont effect other users on the server, which is the main issue why most hosts wouldnt allow it.

Dan