Could you provide us a screenshot about the window of password?
As I know, it needs no special permission to update Teams client.
You could check if your profile is added in Administrators groups as below:
In fact, Teams desktop client updates itself automatically. Teams checks for updates every few hours behind the scenes, downloads it, and then waits for the computer to be idle before silently installing the update. For more details about the update of Teams, you can read this article: https://learn.microsoft.com/en-us/microsoftteams/teams-client-update.
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