yes the REST API is supported in the Remedy OnDemand environment.
You will need to raise a request with the Remedy OnDemand Operations team, through the i.onbmc.com support portal to expose the API.
You will need to have access to a Remedy OnDemand Development environment if you want to develop and test an integration through the REST API.
The REST API itself is documented here:
BMC OnDemand Business Relationship Manager
I pasted the link to our docs.bmc.com site in the response.
If that link does not work, it is now in the subject of this mail and here expanded: