Badges? We don’t need no stinking badges! (Quoted from The Treasure of the Sierra Madre.)
Seriously, though, while badges might be a good idea, there is no official badge to designate a small business.
However, there is nothing stopping you from proclaiming that you are proud of being a small business person on your social media profiles. Or sharing small business memes. Or offering your own discounts to customers that week to thank them for their support.
Celebrate Small Business Week however you wish!