Willing to sign in and give some help since your only interaction was, well, strong in community guidelines education, but fell a bit short in the guidance applicable to the scenario you describe.

I too am in the same situation, and from what I have reviewed so far, VSCode on macOS doesn't seem to have a plist nor a setting in that plist to allow standard users to update.

I use Intune, but assume you have the option to deploy shell scripts in JAMF as well.
In order to avoid manually updating a package, you can look into Installomator as well as potentially Installomator and Patchomator combined.

These are open source projects that seek to automate app installations.
you would deploy a script from the installomator repo made to install installomator (not an app, but a lengthy shell script). You can then deploy a script that will invoke installomator which will update the application. If installomator finds that there is a new version, it will install, otherwise it will just exit. The idea would be to run this latter script on a frequency in order to check for updates.

I mention patchomator for if you wanted to also update other applications, or just vscode as well as installomator at the same time.