Hi,
The index automatically updates, whenever a post is updated. It is possible to set the index to extend periodically, but that will not help this case – as you want to update the specific post.
The best solution would be to trigger the save_post hook programmatically (on the post where the shortcode is used) when the user updates the profile field, and the plugin will automatically update the corresponding post.
Is this a single post, or are there many posts for each user somehow?