Just found these instructions which got me up and running:
https://hostiso.com/how-to-install-setup-node-js/
This will install node and npm in your local account, so nothing needs to be installed globally on the server. node is a compiled binary, while npm is a JavaScipt script that runs with node as its interpreter.
The latest code is downloaded as an archive from here: https://nodejs.org/en/download/ then expanded to a directory in the account.
The binary node is copied from the extracted archive to the bin directory in the account. Now, I found that the bin directory (even though it did not exist until I created it) was already in my PATH. I guess that is a WHM or cPanel feature.
After copying the node binary, the npm script and a number of its required siblings and support scripts are symbolically linked into the bin directory. This means the extracted node archive must not be moved, so make sure it is in an appropriate page. I extracted it to nodejs/ in the home directory of the domain account.
And that was it. Both node and npm are then available as commands when I log in, and that is what Laravel needs to do its asset management, downloading, compiling, minimising, etc.
I have kept the description to just the concepts above, and left the detailed commands to the page I linked at the top. It's all pretty simple in the end, just four or five commands. Updating the commands will just involve repeating the same steps, which take just a couple of minutes.
Of course, laravel will then add layers to that, with its own npm packages that run their own commands (interpreted by node), but that's another level. Getting node and npm was the task here, just giving Laravel what it needs to do its thing.