There are a few different ways to get Chrome extensions installed for users with User Profile disks. Below I describe three of them.
1. Creating a group policy
One method requires a new folder on the file server and a group policy which changes the location for a users %temp% and %tmp% locations. This is needed because UPD's create a junction which chrome cannot write to when it tries to access the users %AppData%\temp folder.
Create a new folder anywhere you would like on the session servers, I'll use C:\temp as the example location. You will need this folder path for the next few steps.
Once the folders have been created, on the domain controller open the Group Policy Management Editor and create a new GPO and name it whatever you would like it to be.
Right click the newly created GPO and click Edit..., this will open the GPO for editing the settings
Browse to User Configuration > Preferences > Windows Settings and right click Folders and select New > Folder
In the Path: type in the path of the folder you just created and add %username%, in my case it looks like this C:\temp\%username%
This is done so when a user logs in a folder will be created for each user under the folder you created on the session servers.
While still editing the same GPO browse to User Configuration > Preferences > Windows Settings and right click on Environment and select New > Environmental Variable
In the New Environmental Properties window select User Variable, in Name: type in TEMP and in Value: type in the path of the folder you just created in the previous step, for me it was C:\temp\%username% and click OK
Repeat this step and use TMP in the Name: field.
The GP should then be linked to the users' OU.
2. Creating a batch file
If you don't want to create a GPO you can do the same as above with a batch file. The batch file should contain the following:
"md C:\Temp\%USERNAME%
setx TEMP C:\Temp\%USERNAME%
setx TMP C:\Temp\%USERNAME%
This batch file will create a user folder in a specified location and set both environmental variables to that location.
2. Creating a Chrome shortcut
One other method would be to create a Chrome Shortcut with the following Target:
"C:\Program Files (x86)\Google\Chrome\Application\chrome.exe" --allow-file-access-from-files --no-sandbox --disk-cache-dir="c:\temp\ChromeCache" --user-data-dir=c:\temp\chromeDataDir
This should launch Chrome and point it to new temp folders the system will create. This will only affect Chrome so other applications will not have to worry about the %temp% folder changing locations.