@Sydney Bruce , Based on my understanding of your question description, firstly to provide clarity on two features/concepts:
Free App Service plan vs free Azure subscription: An App Service plan (ASP) is a way to define a set of compute resources for hosting web apps, mobile app backends, and RESTful APIs in Azure. When you create an Azure web app, you must choose an ASP to host it on. The ASP determines the location, size, and features of the compute resources that will be used to host your app (to scale your apps up or down as needed)
The free subscription, on the other hand, provides a limited amount of resources across a range of Azure services, including virtual machines, storage, WebApps etc. Azure free account FAQ
Short answer:
If you upgrade from Free ASP to Basic, Standard or Premium etc, the quota or compute capacity/associated features are automatically increased.
If you are looking for upgrading to Pay-as-you-go subscription from free, you can directly upgrade from the Azure portal, and the quotas/compute would be upgraded as per the select trier ( you do not need to contact support).
Check out this doc for more info: Upgrade your Azure free account or Azure for Students Starter account
Note: That some quotas may have hard limits that cannot be increased by upgrading. In these cases, you may need to contact support to request an increase in the quota:
You may check that list here: Azure subscription and service limits, quotas, and constraints
I have added more info in the subsequent post/comment as long Answer.
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