From what I've read, a lot of what we think of as being "Nazi obsessions" were really the idiosyncratic interests of individual high-ranking Nazis, not neccessarily shared by the movement as a whole, eg. Hitler himself had a hard-on for Wagner, but the Nazi regime did little to promote his music once in power(and in fact, performances of his music declined). Similarly, Himmler and Rosenberg were into Nordic mythology and occultic esoterica, but other Nazis(including Hitler) found these pursuits laughable.
So, probably the best way to get actual "grammar Nazis" into power would be to have grammar be the obsession of one particular high-level Nazi. I'm not sure who that could be. But there were, famously, a number of artists and writers in the upper echelons of Nazism, and they did generally tend to oppose experimentation in the arts. So it probably wouldn't be too tall an order to have one of them become obsessed with "degenerate grammar" or some such.
That said, from a pragmatic perspective, having to police grammatical minutiae in the workforce and bureucracy might get in the way of the Nazis' broader mobilization goals. So, they might just end up making a few big announcements about it, but then just let the campaign fall by the wayside.