There is a way to a suggest a user follows you after they tweet content from your site. I've seen this done on some sites.
Is there a way to enable this functionality using AddThis?
Yes, add the tw:related attribute to the button, like this:
<a class="addthis_button_tweet" tw:related="YOUR TWITTER USERNAME"></a>
Replace YOUR TWITTER USERNAME with the username you'd like to have people follow after they share. You can separate multiple usernames with commas.