There aren't any cmdlets in the AZ modules to get key-value pairs from Azure app configuration. Are there any other options besides AZ CLI? Is there a way to interact with service from PowerShell?
Take a look at the Az.AppConfiguration module. I can't vouch for the quality of it or how complete it is, but it's out there. The Az module doesn't look like it is included as a dependency.
You can download is from the PowerShell Gallery with:
Install-Module -Name Az.AppConfiguration -Repository PSGallery -Scope CurrentUser