Not really, as that is an OS-specific behavior. Sending a Web API Notification may or may not turn the screen on to display the native notification message. This will depend on a matrix of OS and Browser choices, and won't be reliable. Not the mention the overhead of actually setting the notification channel up on the browser, server, managing subscriptions and clients, etc.
There are tricks to keep the screen on, though, that should work while your page is visible (active tab, in view, not minimized, etc.) Check out this library that achieves that by tricking the browser into thinking that a video is being played: https://github.com/richtr/NoSleep.js