Are modal verbs, such as must and can, considered lexical or grammatical categories?
I think I must be interpreting the question differently from Greg Lee, because my answer is that (at least in English) they must be a grammatical category, because they are different in syntax from other verbs. Of course the individual modals are defined lexically, but what allows us to use a collective term like "modal verb" is grammatical.
Historically, the typical modals form a grammatical category known as Präteritopräsentia: They have past tense form, but present tense meaning. Even in Modern English the lack of an -s ending in the third person singular is a remainder of this inheritance. The verb "will" is historically in a category of its own, but joined the other modals in the Germanic languages.