Yes, a badge can be allocated for a specific team by going to Admin - Rewards and Badges – Click Badges – Click New Badge – Click Teams and select the team you want to assign the badge
You can associate a badge to a reward by going to Admin - Rewards and Badges Menu – Click Rewards tab – Click New Reward – Enter reward Name – Assign Rewards by Badge – Select teams – Choose reward icon or you can browse to upload a custom image – ...