It is easiest to implement a direct send form data to pop-up immediately to the service distribution. All normal services for the mailings has an API and documentation with a description of the methods, and some services for mailing a generator of popups. The advantage of this method is that the service for mailing lists you can configure any series of letters, sent immediately after subscription (i.e., send data from the form in SERVIS), including a letter with a big "unsubscribe" button. Plus you can then API to collect subscriber data in other systems, for example in CRM.
Or you can do the opposite - to first send the data from the form on the website in CRM and then from CRM to service for mailings via the API.
If you go for the first option, I recommend a service mailganer
, it has vstrenny generator popups with a flexible rule setup form is displayed to site visitors. All the data from the form are immediately sent to a list of subscribers, you can configure automatic sending of emails after subscribing.