Okay, so it turns out this is a lot easier than I was imagining. I'll share what I found here in case anyone else wants to do something similar in the future and stumbles on this post.
I'm sure there are a lot of ways to accomplish this, but what I've found is Ajax and the XMLHttpRequest Object.
This tutorial helped me get going:
https://www.w3schools.com/xml/ajax_xmlhttprequest_create.aspSo basically all I need is an HTML page that includes a simple script to read in data from a file and update a segment of the page periodically. It doesnt need to be on a web server, can just be a local file.
VoiceAttack can write out the values I want to show on the overlay to the PitGirl_data.txt file. The next cycle when the script loads the file, it appears on the overlay.
What a time to be alive!