Look for 'subscription form' or 'subscription widget' or even 'lead capture'

https://documentation.mailjet.com/hc/en-us/articles/36004250...

https://sendgrid.com/blog/building-a-sendgrid-subscription-w...

https://gleam.io/app/capture/mailjet

The email addresses end up in their database but you can download the list there. They make money from people sending regular newsletters, that's optional though.