To sync projects we just need the authenticated user to have Admin access to a few particular tools (Highlighted below) on the projects. It includes an Admin tool as well as Documents, Photos, RFIs.
If you are using a specific Project Permissions Template for all of your projects, then please ensure that access to the below-highlighted tools has been given in that template.

So for all the projects that you will configure, the authenticated user needs to have Admin access to those projects in Procore. Your Admin access on those projects will ensure smooth syncing of files and folders between Procore to OneDrive.