I have a non WordPress website in which I'd like users to submit their email addresses. Is there a website widget that I can use to plug this in quickly?
If you just want to collect the email addresses you can create forms using services like JotForm or WuFoo and embed the forms into any website. The forms can include multiple fields of any type (including email address). Both services collect the data for you and you can get to it by logging to your account there or you can receive an email with the responses. Both have free starter editions.
On the other hand, if you are collecting email addresses so you can send out a mass email to those people who sign up, then services like MailChimp or iContact are more what you are looking for. They not only collect the email address, but also allow you to design/write an email and then do all of the mass sending for you.