B Corporation is a private certification affirming that a certain company has achieved "the highest standards of verified social and environmental performance, public transparency, and legal accountability to balance profit and purpose." It is issued by B Lab, a non-profit organization. These are the people involved in issuing a B Corporation certification. Just to be completely clear, you wouldn't be "filing" to be a B Corporation the way you would file to become organized as an LLC with your state's Division of Corporations. You would simply be applying to a private sector body for a certification of your corporate practices.
Therefore, a B Corporation certification can be held by any for-profit company of any kind of business structure. You can be a B Corporation-certified LLC, LLP, sole proprietor, corporation, and so forth.
You can also be a B Corp-certified S corporation, or a B Corp-certified C corporation.
This is just a footnote, since what I'm about to explain next doesn't really answer your question, but provides a useful clarification. In the United States, companies are registered with a state's Division of Corporations (or equivalent), but are also liable for federal income tax. Since different states offer different business structures, the federal government needs to classify all these structures into their own defined categories for the purposes of federal tax calculation.
A C corporation is recognized by the federal government to be any business that is taxed separately from its owners. In other words, if I were the shareholder of Apple, a C corporation, I would have to pay my personal taxes on dividends earned from Apple, while Apple has to pay corporate income taxes on its profits, regardless of the (taxed) dividends it has paid out to me. This leads to what is called double taxation.
Corporations and LLCs would commonly be regarded by the federal government as a C corporation.
An S corporation is recognized by the federal government to pass corporate income on to shareholders, who then pay personal income taxes on corporate income. Therefore, tax on corporate income is only paid once.
Sole proprietorships would commonly be regarded by the federal government as an S corporation.
Just as a concluding thought exercise: a firm can be organized as an LLC under state law, certified as a B Corporation, and recognized by the federal government as a C corporation. It can actually be all of these three things at once.