On Windows, you can cancel the admin prompt when you install Zotero, and it will perform a local installation that you will have permission to update as a local user.
We also provide a ZIP version of Zotero for Windows that you can extract and run from any folder. The ZIP version doesn't automatically register Zotero as a handler for various file types (e.g., RIS files), so we recommend using the installer if possible.
On macOS and Linux, you can run the application from any folder (e.g., an Applications folder inside your home folder on macOS instead of /Applications).