So basically what you need is a way to integrate a way for people to purchase the subscription with something that will automatically add them to a list.
There are some paid options for this, but I think the cheapest way would be to set up a store with Wordpress and Woocommerce to
sell the subscription (it doesn't need to be a full store, but just a way for people to buy the subscription) and set up an integration with zapier. Zapier is an
online service that connects different apps/services so you can set it up so that it automatically grabs new buyers and adds them to an email list to Mailchimp or whatever list provider you use.
I would recommend Mailchimp since it's a decent service and is free to start with, which is nice. I started with Mailchimp and switched when I outgrew their free subscription since I needed a bit more than what they offered for my business but for what you're talking about I think it would work really well.
Wordpress and Woocommerce are free and there are lots of free themes out there and whatnot. You may need to buy an SSL certificate to make the checkout secure, though, don't quote me on that since I'm not sure.
You CAN do the same thing with Paypal and Zapier instead, if that makes it easier, but Paypal is a "premium" integration with Zapier so you'd need to have a paid Zapier plan. I think you could do it cheap or nearly free with the Wordpress/woocommerce combined with Mailchimp and Zapier to connect them.