I found this guide that states custom metadata is available in Professional, Enterprise, Performance, Unlimited, and Developer.
According to the developer documenation:
To use Metadata API, your organization must use Enterprise Edition, Unlimited Edition, Performance Edition, or Developer Edition.
It does look like you can request Metadata API access for Professional Edition:
ISV partners can request Metadata API access to Professional Edition orgs for apps that have passed the AppExchange Security Review. Access is granted through an API token (client ID). This special key enables the app to make Metadata API calls to customers’ Professional Edition orgs.