Hi Mikediscre,
It sounds like you maybe going to some type of website to approve posts, but you don't see post requests when using Safari, only in FireFox.
To troubleshoot Safari when a website is not working as expected, you can follow If Safari on Mac doesn't open a webpage or isn’t working as expected.
All of the steps are good to follow. This one particular step can be very helpful for this type of issue:
Test with a private window
A website can store cookies, caches, and other data on your Mac, and issues with that data can affect your use of the website. To prevent the website from using that data, view it in a private window. From the menu bar in Safari, choose File > New Private Window, or press Shift-Command-N.
If that works, use the following steps to remove the website's data, including its caches and cookies. The website can then create new data as needed. If it's a website that you sign in to, make sure that you know your sign-in information before continuing.
1. Choose Safari > Preferences, then click Privacy.
2. Click Manage Website Data.
3. Select the affected website from the list shown.
4. Click Remove.
5. Click Done.
6. Open the website again in a non-private browser window.
Ultimately, if Safari is working for everything else, you may want to go to the developer to assistance such as to see if Safari is compatible with this, or if there are any settings or add-ons you need when using Safari.
Take Care.