It's a ridiculous name, of course. Henry Ford was a person, and even an uppercased the would do little to alert the reader that such a minimalist reference to the person's name could mean anything but the person himself. But they went and made the moniker even sillier and more inscrutable by lowercasing the the, as though every printed reference would be rendered in the typefaces and colors of the logo.
Those of us who make style decisions are used to slapping down capitalized thes and even eliminating the word altogether (a lot of us lowercase the the even in references to newspaper names in which it is uppercased, and The Ohio State University, to any sensible editor, is Ohio State University). For many publications, then, the people of "the Henry Ford" have made things easy. For publications that honor the Thes, however, there is an intriguing conundrum. When you cap the The in The New York Times, you're really doing so not because the word is capped in the flag, but because the word is present in the flag. We capitalize names, and if we are recognizing The as an integral part of the name, there you go. (We would write "Los Angeles Times" even if the flag said "los angeles Times.")
So, if you humor the NYT people, does it follow that you should ignore the wishes of the Henry Ford people and write The Henry Ford? I say yes.