Automated Building

Is there a way to implement a build process for Roku - Jenkins integration, build scripting, etc?

Sure, if that rocks your boat.
I don’t see the point, since all a “build” phase for Roku does is create zip bundle of the needed files (there is no pre-compiling) and what a “deploy” does is to http upload that to the player, which as side effect also "run"s the app. I just use half-dozen-line script to build-deploy-run from my text editor, it is fast enough i think of it a simply “run”.

What would that script look like, and is there any documentation on it?

My personal script is here viewtopic.php?f=34&t=70133#p442404

The case with the makefile by RokuCo is more complicated - viewtopic.php?f=34&t=65779 - but i have hopes that at some finite point of time RokuCo will fix that.

For all practical purposes both do the same.