As others have said, "home delivery" is far more common, and "home" as an adverb is generally used other than in "home delivers". That said, "home delivers" is immediately understandable, and probably most people would read it in context without batting an eyelid.
I suspect that if you asked people what "home delivery" usually refers to, they would name newspapers, furniture, groceries, pizza, and the like. A serious pharmaceuticals company may want to disassociate from more trivial kinds of home shopping/home delivery and thus choose a less cliched, less "fun" sounding term. This is entirely a stylistic choice. It should be weighed against what is standard, but, if there is no effect at the expense of comprehension, it is not an invalid choice.