The current version of ChromeOS will run Debian Linux apps: https://techcrunch.com/2018/05/08/you-can-now-run-linux-apps-on-chrome-os/
Upwork Time Tracker has a version for Debian: https://www.upwork.com/ab/downloads/os/linux/
But, the download link won't appear on a ChromeOS browser. The workaround is to access the download link from another computer that will identify as a Linux computer (like a Mac), and then the link will be available to you. Once downloaded, make the installer program available from Google Drive, Dropbox, Sharepoint or similar so that you can download it from your ChromeOS device.
Once the installer is on your ChromeOS device, move it from My files/Downloads to My files/Linux files. Right-click on it from there, and the pop-up menu will present the option to install. Once installed, the Time Tracker icon gets added to your list of programs. (You can then delete the installer from your disk.)
The workaround for ChromeOS is not officially supported by Upwork. I hope it works for you. (I'm personally not able to get screenshots to work on ChromeOS, yet.)
It would be really nice to have a native chrome tracker. And it would give additional advantages as a seamless upgrade. Personally I hate the way how it does in Ubuntu.
Also, I use Cloud9 for development to not be stick to a PC. So I do not need any OS, just Chrome and Upwork tracker.
So thank you, Brandy, for the raising the question
Right!! This is driving me crazy. I hate Windows, but my new gig requires the tracker. Sometimes I have to re-upload it to the dumb laptop. For almost three years I'ved worked exclusively on my Chromebook. Toggl has a extension and it's so easy. Why doesn't Upwork???