Hello Andy Cheetham
Thanks for your question.
This issue may point to RBAC rather than the security group itself. Security groups usually control network access while RBAC controls Role based Access. You'll need to ensure they have the necessary permissions to access and read the telemetry data collected by Application Insights
These can be found in the documentation here: https://learn.microsoft.com/en-us/azure/azure-monitor/app/resources-roles-access-control.
For least privilege I would recommend you try the Monitoring Reader role firstly and then if that doesn't suit your needs you can go upwards.
See Snippet of the documentation here:
Please let me know if you have further questions
You can mark it 'Accept Answer' if this helped.