Hello @Meghan and welcome to the Admin hub!
My strongest recommendation is to keep it simple.
Create a basic Google Form where you ask the staff user who wants to use a new app, extension or service to provide you with as much details as necessary, including links to all documents.
Name, link to creator's website, link to actual app/service/extension, link to Terms of Service (ToS), link to Privacy Policy (PP) and link to Data Privacy Agreement (DPA), where the thing will be used (class year, subject, school, whatever you need), and, very important, WHY should it be allowed (especially, if you already have similar things already allowed!).
Then you need to have a serious discussion with your boss, and/or the boss' boss and the legal team at your school/district, and write down the ground rules for what is required to allow a new thing, and who is going to be part of the group of people doing the vetting.
THAT IS THEIR RESPONSIBILITY, and not yours! Your job is to click the necessary buttons to allow something, and be a technical support for the people responsible for conducting the actual vetting!
I must also recommend that you publish the list of applications and their result, so people don't apply for the same things. I've used Looker Studio to create a dashboard (makes it fairly nice and easy to use) based on the Responses Sheet from the Form. that way people can check there before they even start filling in a new application.
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https://wheretofind.me/@NoSubstitute