Hello Frank Lunenburg,
Greetings! Welcome to Microsoft Q&A forum.
I understand that you’re encountering issues with the Azure Cognitive Services API when restricting access to your blob storage.
When deploying the Custom Vision API in a limited private network, configure your virtual network to allow access to the Custom Vision service -refer https://learn.microsoft.com/en-us/azure/ai-services/cognitive-services-virtual-networks?tabs=portal#supported-regions-and-service-offerings, https://learn.microsoft.com/en-us/rest/api/custom-vision/?view=rest-computervision-v3.1. for detailed guidance. Ensure that your application can reach the Custom Vision endpoint from within the network.
Verify that the networking settings allow traffic from your application’s location (IP) to the blob storage. Review any firewall rules or network security groups that might affect outbound requests from your application.
Furthermore, I wonder if you have configured these other settings as well:
Similar SO thread for reference - https://stackoverflow.com/questions/77105932/azure-cognitive-search-on-private-endpoints-implement-function-for-built-in-sk ,https://github.com/Azure-Samples/cognitive-services-vision-csharp-sdk-quickstarts/blob/master/CustomVision/README.md
I hope you can find this useful to overcome your concern. Please let us know if you have any further queries. I’m happy to assist you further.