We wanted to use this as a T-shirt quote, but I feel that "if winning isn't everything then why do they keep score" is wrong. The correct sentence should be "if winning isn't everything then why do they keep a score". However, a Google search shows that I may be wrong. Only one person agrees with my sentence. Is the first sentence just incorrect American grammar?
"Keep score" is a (slightly) idiomatic expression that means "keeping track of the score". It is not ungrammatical to use this phrase on a T-shirt (or anywhere else). If the T-shirt had the text "keep a score", it might actually give the impression that the shirt was not made by a native English speaker.
(This is speaking as someone from the US; I don't know for certain if the expression might be different in the UK.)
You'll want to add the apostrophe to isn't and a question mark to the end, but there's really nothing wrong with putting such a sentence on a T-shirt. T-shirts and bumper stickers are pretty much exempt from formal rules, and no one expects them to be models of English grammar and usage anyway. I wouldn't worry about pleasing the editors on this one.
Casual English is appropriate for a t-shirt since wearing clothing with phases on them is pretty casual. The term "keeping score" can have the meaning to keep track, but may a negative connotation of holding grudges.