Add Existing Users to Jetpack Subscriber list?

You can enter different email addresses in the subscription form on your site, thus triggering a confirmation email to be sent to all these email addresses.

You cannot, however, add subscribers in bulk, or add them without their consent through the confirmation email.

To make it easy for your existing users to subscribe using Jetpack Subscriptions, I would suggest the following:

1. Create a new Page on your site, named “Subscribe”
2. Insert the following shortcode in the page content:
   [jetpack_subscription_form]
3. Send a link to that page to all your registered users, to allow them to subscribe using Jetpack Subscriptions.

Or is there some other way to do this within WP (on my own website)?

It might be worth searching the plugin repository for plugins allowing you to email all registered users at once.