I'm currently unable to update the software to the latest version, since any attempt to do so will result in the following error page:

All third-party applications and plugins are currently disabled.

I'm currently running v4.5.4.2

Download the IPB files from your client area and manually upload them to your website. Then run yourdomain.com/admin/upgrade. 

I woudl also suggest temporarily disabling varnish cache while you are running the upgrade. The error code there indicates the server was unavailable for the request at the time it was run

Updating php to version 7.4 seemingly solved the problem. I am a little worried about version 8.0 because when I tried to set it up it returned error 503 on every page of the site, even those not related to the update.

1 minute ago, Shiyo said:

Updating php to version 7.4 seemingly solved the problem. I am a little worried about version 8.0 because when I tried to set it up it returned error 503 on every page of the site, even those not related to the update.

You'll want to check PHP 8 with our compatibility checker for our basic software compatibility: https://invisioncommunity.com/files/file/7046-invision-community-requirements-checker/

However, if everything is failing, working with your hosting provider, they should be able to outline the issue for you.