I'm having the same issue, and I have used an access token from my Shopify's private app. Both the Admin API and the Storefront API access keys result in a 403 when used with Azure's connector. Both access keys I've confirmed do work normally. There is no information available about what the Azure connector is connecting to.
The azure data connector does not state which API it uses to connect to Shopify. The lack of basic info is egregious. Please find the L4 developer who deployed this halfbaked crap, contact his manager, and let the manager know that their incompetent subordinate should be placed on a performance improvement plan.