First, let's define our terms. An umlaut is a diacritic mark that changes the roundness of articulation of the vowels it is used with: /a/ /e/ /o/ and /u/. The word is a combination of German "um," meaning "about" (as in to come about; change) with "Laut," meaning "sound." Umlauts and all diacritic marks change the articulatory sound or emphasis of the letters they mark and modify.
Grammar is the sets of prescribed rules by which languages produce consistency, quality, and meaning in how they are spoken and written. One rule of English grammar is that an Object is associated with the Verb and usually follows the Verb: The boy gave a whistle. [For certain rhetorical or poetical effects, the Object may come ahead of the Verb: A whistle gave the boy.]
Now let's examine these two together to find the answer to your question about the usage of the umlaut in grammar. If grammar governs rules of word order and sentence construction, and if the umlaut, like all diacritics, governs the quality of articulation of sounds (pronunciation), then the umlaut can be said to have no function and, therefore, no usage in grammar. This is because grammar and articulatory diacritics (like the umlaut) have no relation to each other.
The one exception to this is if an umlaut changes the word class of a word in a language other than English. In this case, the relationship between the umlaut and grammar would be that it may change word class and thus change the word's function in grammatical constructions.