DM gave my party member a new magic item read as follows: 

Leather Armor of Advantage. +2 Leather Armor. Once per turn, you can gain advantage on your next attack. Sneak attack deals an extra 2d8 damage.

I'm the guy who implements the homebrew items into DDB best as possible.

Is there a way to apply the bonus sneak attack damage to a character sheet using the homebrew item builder?

No, unfortunately there is not. They will have to make a custom action on their character sheet to roll the additional 2d8, or just roll it from the tray.