You do not need to host email on the same system as your web hosting. There are many ways you can send email via the website using remote mail servers.
For example I use Google Apps (Gmail) on my server that is hosted by another company, I have no problems sending out emails and receiving emails via contact forms.
You've not mentioned much about your web host which is the key element here for sending out emails to your visitors. I'm assuming that you have Linux since most hosting does, then you should have a thing called sendmail on the server you can opt to send emails with or without authentication - in other words you don't even need to tell it the POP3/IMAP settings as it can send emails out without this option, however better success rates (Junk Mail Triggers) are achieved by using an STMP sender.
Most businesses and sites require the sending option of emails which you don't actually need an email account to do, so it doesn't matter where your email is hosted. Contact your host and ask them to provide you with the contact.php or information on the send mail they use.