How to allow Contributors to publish articles after approval

How can I make the contributor rank able to publish an article but
have it then require the approval of an admin before it actually gets
published to the website?

The “Contributor” role already behaves that way:

Contributor

  • delete_posts
  • edit_posts
  • read

https://codex.wordpress.org/Roles_and_Capabilities#Contributor

As you see, that role lacks the publish_posts capability of the role just above it– the “Author” role. Without that capability the “Contributor” can’t publish but can only save drafts and edit.