Usually, the HTTP 401 is a response related to an issue in the authentication process in the code due to invalid, missing or expired tokens.

Codes shared by @guillaume has similar logic from the official doc / GitHub and should work (but I guess not in this case).

Below steps / alternatives might be worth double checking:

• Ensure that the service account has the cloud run invoker role

• Apply troubleshooting:

  • Make sure that your requests include an Authorization: Bearer ID_TOKEN header, and that the token is an ID token, not an access or refresh token.

  • Redeploy your function to allow unauthenticated invocations if this is supported by your organization. This is useful for testing.

• Explore generating tokens manually

Could you share the link referencing that Serverless VPC Access connector is a potential cause?

As a last resort, you can reach out to the paid support for detailed troubleshooting of the issue with Cloud Run functions specialist.