Without using the mouse, how do I trigger 'Print' in Gmail (using Google Chrome on the Mac)?
Here is one way that I triggered the Print sheet in Gmail within Google Chrome using a keyboard shortcut:

Doing this on macOS High Sierra1, in Automator, create a new Service.1

Set Service receives1 no input in Google Chrome

Add a Run AppleScript action, replacing the default code with the following example AppleScript code:

tell application "Google Chrome" to execute front window's active tab ¬
    javascript "document.getElementsByClassName('T-I J-J5-Ji T-I-JN L3')[2].click();"


Save the Service as, e.g.: Google Chrome Gmail Print

1 In macOS Mojave, and later, an Automator Service is called a Quick Action. There are also other minor nomenclature differences but they should be more obvious when comparing them to pre-macOS Mojave Automator workflows.



Assign the Service a keyboard shortcut in: System Preferences > Keyboard > Shortcuts > Services

I assigned ⌃⌘P to the e.g. Google Chrome Gmail Print Service.

Now in Gmail, within Google Chrome, having an email opened showing the print icon, as shown circled in red in the first image below, and I press ⌃⌘P the Gmail Print sheet, as shown in the second image below, is revealed.





Note that in the standard view, as shown in the Gmail image above, there are four ElementsByClassName with T-I J-J5-Ji T-I-JN L3 as the class name. Index [0] and [1] are used by the Collapse all icon or Expand all icon that will show, when applicable, to the left of the Print all icon [2] and index [3] is used for the In new window icon to the right of the Print all icon.





 Note: The example AppleScript code is just that and does not contain any error handling as may be appropriate. The onus is upon the user to add any error handling as may be appropriate, needed or wanted. Have a look at the try statement and error statement in the AppleScript Language Guide. See also, Working with Errors.