Is it possible to use Azure Data Factory to get data from a REST API and insert it to a Azure database table?
I have done this using Custom .Net Activities. I had a need to pull data from Salesforce API. I have a write up on how to do this here: http://eatcodelive.com/2016/02/06/accessing-azure-data-lake-store-from-an-azure-data-factory-custom-net-activity/ it is utilizing the data lake store but you can store the data anywhere.
Also have a look here for an example from microsoft: https://azure.microsoft.com/en-us/documentation/articles/data-factory-use-custom-activities/
Hope that helps.