Your routers wouldn't speak HTTP. Your web service endpoint would talk to your network management middleware, which would then issue your respective IOS/NXOS/JunOS commands to your core or edge gear.

You'd grant your customers the ability to null route traffic from IP blocks (/24 or larger, because ain't nobody got memory to route blocks smaller than that in IPv4) so they wouldn't saturate their links with useless traffic. There was a discussion on the North American Network Operators Group (NANOG) mailing list a few weeks ago.

Disclaimer: I have operated large-scale networks for over a decade.