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Ah, that is a gap in our documentation. There is indeed a REST API for creating a stack it is just missing from the docs. I opened a bug so we can backfill the docs. In the meantime, the API is:

POST /api/stacks/{organization}/{project}

And here are the types for the payload: https://github.com/pulumi/pulumi/blob/master/sdk/go/common/apitype/stacks.go#L45-L54 A post body along these lines is the minimum:

{
  "stackName": "yourStackName"
}

Ooo lemme try. Thank you!!