In the 0.8.7 update in order to remove the older installation of Modrinth App, the installer has been switched back to require Administrator account access. This unfortunately resulted in update failing for users on Windows system that use account without administrator level.
To resolve this error, follow these steps:
Uninstall existing Modrinth App versions through Windows Apps and Features settings.
Download Modrinth App from the official website.
Run the installer as Administrator and install Modrinth App normally.