I am also facing exactly the same issue.
My API works from Web activity but doesn't work from the Rest Datasource of Copy Activity.
It was working initially and after a week when I tried it suddenly it fails with error 406. If I add Http Json data source adding additional headers Content type, Ocp-Apim-Subscription-Key and Authorization with Bearer token is so difficult to format it doesn't work too.
Json data from web activity added separately is also difficult to format and pass to azure sql table.
Failure happened on 'Source' side. ErrorCode=RestCallFailedWithClientError,'Type=Microsoft.DataTransfer.Common.Shared.HybridDeliveryException,Message=Rest call failed with client error, status code 406 NotAcceptable, please check your activity settings.
Request URL: 8***/.
Response: ,Source=Microsoft.DataTransfer.ClientLibrary,'