Currently the only way to run a web tool programmatically outside of ArcGIS is to create a root-level developer credential and use that to generate a token to pass into the submitJob endpoint of the web tool. I've tried scoping the developer credential to the web tool itself, as well as the Python script from which the web tool was published, and the the submitJob response says a token is required to access the resource. However if I use a root-level token I'm able to successfully run it.
