When attempting to modify or install the GitHub App VCS integration via the organization settings, it may appear greyed out, suffixed with (Installed), as shown below:
The GitHub App entry in the organization's VCS settings page is purely visual to show that it is available. The GitHub App is marked as installed in all organizations by default as it is a globally-installed integration in TFC/TFE that is activated on a per-workspace level.
Once the GitHub App is added to GitHub, all management of it should be accomplished via GitHub. The only configuration is which repositories the app has access to (or to revoke the app's access completely). This can be managed from https://github.com/settings/installations for users or https://github.com/organizations/<organization>/settings/installations for organizations.
Because the GitHub App is enabled per-workspace, if the app has been revoked it can be reauthorized in the workspace settings (or while creating a new workspace). With an existing workspace, visit the workspace's Settings -> Version Control page and if there is already a VCS provider linked, choose "Change source". If there is not a VCS provider linked, choose "Connect to version control". Choosing "Version control workflow" when it asks for the workflow type and choosing "Connect to a different VCS" will show the VCS provider list and choosing "GitHub.com" (this is the GitHub App) under the GitHub dropdown will begin the GitHub app authorization process. After the app is authorized, it can be managed from GitHub again.