Is there a way to sync GeoJSON files to my cartoDB table?
If this is not yet supported, what would be the best way of implementing this feature (syncing the data frequently with the existing cartoDB table) myself?
It might be worthwhile looking at the CartoDB Import API. You can set up syncing a file from a URL with a simple cURL command.
Check out the documentation and the example at this link. It may help get you started down the right track.
While the import API is one possible method, it looks like you can sync GeoJSON if it is hosted on the web somewhere. This method was outlined on the Fulcrum guides and can be seen here, read the section "Creating A Synced CartoDB Table".
Steps are as follows.
I used GeoJSON from Mapbox to test and can be found here