How can I install packages for custom support scripts?

Start by telling us what binaries or data you need and we can give some specific advice on how to get them. There is deliberately no way to install deb/rpm packages in the OS but that's not necessarily the end of the story.

LibreELEC is an intentionally simple distro but we should support all the normal bash scripting commands because we use the same things within the distro. Note that we use busybox for lots of things which can reduce command options. The full /usr/bin/vcgencmd binary exists in the image already.

For things like GPG and Dropbox we don't have any native capabilities and unless you want to build custom images that you added more packages to (do-able but probably more work than you're looking for as we build everything from source) the main escape route will be using Docker (installed as a Kodi add-on) to run a container that provides whatever binary that you're missing. You get bonus credit for authoring your own container that adds all the missing stuff in one container to avoid running loads of containers (overhead you don't need, and the one thing that's likely to result in high temps etc.).

NB: While I understand that some users like to monitor everything, the controlled and appliance-like nature of the distro also largely eliminates the scenarios where general purpose distros drift into bad shape over time requiring monitoring. LE is super lightweight and highly optimised for Pi hardware and with normal movie watching duties the only thing that's needed for avoiding high temp and under-volt scenarios is a half-decent passive or fan-cooled case and the official PSU. I've managed enough infrastructure over time myself to understand the inner need to monitor, but watching movies is more fun