The short answer is yes.
Under the hood, Construct's Gamepad plugin uses the Gamepad API, which does support Switch Joy-Con controllers. You can test your connection using this demo page.
With that said, Joy-Con controllers don't have a D-pad, and you might have mixed results with the analog stick. Construct's Gamepad plugin attempts to remap input from the Standard Gamepad layout to the Xbox 360 controller layout. Therefore, you may be better off using an Xbox 360 controller, PS4-5 controller, Nintendo Pro Controller, etc.