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Unfortunately APIGateway Websocket APIs are not supported yet.  Pulumi exposes the upstream Terraform Resource Provider resources, and this has been pending in the Terraform AWS resource provider for a little over a year :disappointed:.

<https://github.com/terraform-providers/terraform-provider-aws/issues/7004>

Oh, bum. At least I feel a little less stupid.

What would be the quickest way for me, in the future, to determine whether a service I want to use is or isn't supported by Pulumi?

In general -the answer is to look at the docs: <https://www.pulumi.com/docs/reference/pkg/nodejs/pulumi/aws/>

In this particular case, the confusing design of API Gateway (v1 vs. v2 instead of actual named things) in AWS makes it a little less clear what you should be looking for unfortunately.