Upload image to media library using TinyMCE

I am doing my own form (using WP User Form plugin) for my user to submit posts from the front end. For the content text area, I’m using a custom TinyMCE toolbar. I want my users to be able to add images to their posts. Using these instructions from TinyMCE’s docs I’ve managed to get images to show up in the textarea, but when I submit the form the images won’t be uploaded to the Media Library, and won’t show in the post page. How do I also upload the images when submitting the post (or upload them at the time the user adds them, like the built-in Add Media button works).

Thank you!