Do you know what software runs your web site? It is some simple web pages? Maybe it's a site powered by WordPress?
If it's WordPress there are plugins that you or your web developer can install to request email addresses from your visitors.
Do you have a store? A "sign up sheet" in the store might work too but that can lead to typo's in your email address lists.
How do you plan on contacting your coustomers and what kind of content do you plan on sending to "stay in touch"?
If you need a reliable email newsletter service I recommend
MailChimp both for their free sending limits and available templates.