Force Chrome Relaunch After Update

You want  to set the action on startup key  to restore last session. Note this is a security concern in certain industries

I'm not seeing action on startup as an option. Where is that at? 

If forcing a restart isn't best practice is there an option to require a restart before being able to do anything else? 

Try "Files and Processes" pane to executed command.
open -a "/Applications/Google Chrome.app"

If you are using Jamf Mac Apps, there is an option under End User Experience to set a post-update action, and you can tell Jamf to relaunch the application after it finishes updating. If you are using a policy, as @pacificusexpell said, you just need to use files and processes command to launch the app.

You can add in post install script or files and process to run after installed the pkg file, also you can add notification in the script.

I have already added the post update action in the end user experience, but my laptop didn't restart. The update was already installed but I haven't relaunched the browser yet. I wonder if Jamf needs to install the update (instead of Chrome doing it) for the relaunch to trigger. 

The post update action does not restart the computer, it restarts the application if it is already open. If the application is not opened, Jamf just updates the application and moves on.

You may need to call Jamf Customer support for an explanation of how this workflow functions. 

Right, I meant restart my browser not restart my laptop.