Welcome to Microsoft Q&A forum.
As I understand, you want to Update Extension Versions Without Superuser Privileges in Azure Database for PostgreSQL - Flexible Server.
Here is the complete documentation on How to enable and use pgvector
on Azure Database for PostgreSQL - Flexible Server:
https://learn.microsoft.com/en-us/azure/postgresql/flexible-server/how-to-use-pgvector
I am checking internally if there is any workaround without superuser, thanks.