Here is a link to the API documentation..
Scanning through it, it looks like it's more for getting information from and interacting with the puppet master, not for interacting with individual puppet deamons to start off a run.
There used to be something called 'puppet kick' which could do what you were looking for, but it's been deprecated a long time ago and I believe removed as of 4.X IIRC.
The generally preferred orchestration framework is MCollective
I'm not sure if mcollective has a rest API or a client library. There is a section on the documentation about writing a SimpleRPC
I hope this helps point you in the right direction.