I've just figured out how to get PICO-8, sound and all, to work on ChromeOS 79. Since there doesn't seem to be any other information on the topic, I'll post it here.

To begin, install Linux (Beta) (a.k.a. Crostini) on your Chromebook. You can learn how to install Linux (Beta) here.

After Linux (Beta) has installed, install and unzip the Linux version of PICO-8. Then, drag and drop the pico-8 folder into My files / Linux files. That way the files will appear in the home directory of Linux (Beta).

Next, we need to make pico8 executable:

$ cd ~/pico-8
$ chmod +x pico8

That's enough to get PICO-8 working. However, the sound won't work. To fix this, open the new Terminal app and type the following commands:

​​# Make sure the new cros-pulse-config package is installed.
$ sudo apt-get update
$ sudo apt-get dist-upgrade

# Clear out existing pulse settings.
$ rm -rf ~/.config/pulse

# Turn it off & on again via crosh (Ctrl-Alt-T).
crosh> vmc stop termina

Finally, to run PICO-8:

$ cd ~/pico-8
$ ./pico8

And there we go! PICO-8 running on ChromeOS.

Sources

Lexaloffle - A proper Linux installer, please
Chromebook Help - Can't get my Linux apps to produce sound
Chromebook Help - Set up Linux (Beta) on your Chromebook