Add support for building, testing and deploying the API contents via GitHub Actions

[CAM-Gerlach]

Currently, the API contents in the gh-pages branch must be manually rebuilt. Building it in PRs and deploying it on pushes to the main branch will avoid the need to manually perform this convoluted and maintenance-intensive process, as well as document it in the Contributing Guide, and will keep the API content in sync with the code and serve as a basic test of PRs.

If desired, Netlify support could also be added, in order to preview the API output from PRs; my current GH workflows include that as well. The one difficulty there aside from granting app permission for this repo is managing the Netlify account, as it doesn't seamlessly integrate with Github's authentication. I'd either need to run the preview off my own Netlify account, or we could create one specifically for this project and share the credentials with each other. Not the cleanest way of accomplishing things, but not sure of a great alternative. So I suggest saving that for a followup, since it isn't critical.