Click the envelope button in the social/friends tab, enter their Bungie ID in the search field using the virtual keyboard, click the result, and then select "Send Bungie Friend Request" in their profile when it pops up.

So, I was poking around on my Xbox, trying to figure out how to send a Bungie Friend Request to a specific Bungie ID to someone who plays on PlayStation. For the life of me, I couldn't figure it out! I even tried the bungie.net/friends page, but that requires us to be platform friends first...

Is there a menu/setting somewhere in the game that would let me pop up the virtual keyboard to type in someone's Bungie ID and send a friend request directly?

Scroll down to the Manage Invites tab on your Social page:

On the search bar, type their full Bungie ID. If it's recognized, you then just click on the tab again name again, and Send Bungie Friend Request. Once Accepted, you'll see the grippy hand on the right side of their name plate.