I build cornhole (bag) toss tables and sell them. I want to put sports logos on them. Can I legally sell them with trademarked logos?

No.  The St. Louis Cardinals own their various trademarks, including their logo, and if you want to use it for commercial purposes you will need their permission.  It is likely that the Cardinals have already licensed someone else to be the exclusive licensee of their logo on the type of product you want to sell, but even if not, and even if the Cardinals are ok with having their logo associated with your product, they will want you to pay a royalty for using the logo.