You can use HTTP functions to expose an API of your site’s functionality. So basically, you can add code to your external site so that once a user is logged in to the site, you post a message to the backend of your Wix site.
https://www.wix.com/corvid/reference/wix-http-functions.html
Wix Stores code example for you to look over on answer to this reply on StackOverflow:
https://stackoverflow.com/questions/45602997/wix-store-get-list-of-products-from-their-api-to-show-on-mobile-app
Wix Stores API:
https://www.wix.com/corvid/reference/wix-stores.html
https://www.wix.com/corvid/reference/wix-stores-backend.html