How can I reach people from our registered site members who have not bought something and are not on our email list?

It sounds like you have their email address, so I recommend using it to target them on Facebook and Twitter through their custom audiences program. You can upload the lists there and specifically reach these users ( and lookalikes). Twitter even has an ad format that send a targeted email subscription ad with an embedded form.

Your only email option is to send a transactional or relationship message such as a website maintenance, password reset request or terms of service update. As long as the primary purpose is non-commercial, you can include some promotion of the newsletter below the other copy.

You should be able to send an email to people who registered for your site, unless your privacy policy says you won't. There's noting wrong with sending them an email to "update their info" and offering them the newsletter at the same time.

There are a lot of variables but the first thing that comes to my mind is coming up with some offers and CTAs that will allow you to convert them into email signups. Then you should be segmenting your email list based on the core wants, needs, and interest of the members in order to send them targeted offers. Sounds like you have plenty of leads who have opted in already, so it's really a function of reactivating the customers who have already bought something while making lower level value offers to get the rest engaged. Depending on the business this can go a lot of ways, but you have a nice setup already with a good set of leads and members. They just need to be activated with good offers and a solid marketing funnel.

As others have said, if you have their email list you can create an audience through Facebook so that you can use re-targeting ads to get them to register for your newsletter list (assuming there's a specific newsletter list).

You can also set up a retargeting pixel to target everybody on Facebook who's visited your site/URL to target "warmish" through Facebook ads.