There is not an attorney familiar with intellectual property that will not tell you to seriously reconsider using the teams' names without some form of license. That being said, it is not automatically out of bounds to use the names. Rather, a license to use those names in connection with the product you have in mind would have to be inked that clearly states that you have the rights and the parameters (royalty fees, scope of use, etc.) that both parties would have to adhere to. A possible course of action for you would be to approach one team at a time and determine who handles its licensing affairs. That office/party may ask for a proposal or other materials to get the ball rolling. And you may need the assistance of an attorney familiar with merchandising and licensing.
I hope this helps.
Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.
Yes, it's almost certainly a violation of those teams' rights. Before you start on online business involving intellectual property like copyrighted works or trademarks, or any business, do some reading on starting a business and how IP rights are acquired and protected. Nolo publishes very good guides for non-lawyers, and you can also check Amazon or your local library.
Then once you know what you wanbt to do and your research indicates that it's a viable business, consult an IP/business lawyer to help you form a business entity, choose a business name and tradename, acquire a domain name, acquire necessary insurance and licenses, formulate a product line and marketing/advertsiing plan, and license or acquire the appropriate IP rights.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.