API Web API/Pages are possible to create in ORIGAM Architect at Model Browser as shown on following picture.
If you have created some API, you have to add OrigamPagesModule into your web.config.
If you plan to use PUT/DELETE functionality for your API, you have to set
runAllManagedModulesForAllRequests=true (see above), because PUT and DELETE HTTP requests are not considered by IIS as managed and therefore a request is not processed by .NET (managed) pipepline. Otherwise it wouldn't e.g. call form authentication for such a request.
Furthermore, you have to disable WebDav module, because it handles PUTs and DELETEs sooner (If WebDav is configured).