In terms of grammar, there isn't much of a variation in what you can change.
In it's simplest form, a language needs to have words that refer to things, (such as objects, people etc.), and also words to describe what that object is currently doing. Extrapolate this to all other variations of 'things to refer to' (emotions for example), and the most variation you can have is what already varies in most languages on earth.
However, alien life may communicate in many different ways. They may whistle at different pitches, shake their bodies in specific movements as a form of sign language, or have a bio-luminescent antenna that brightens and dims in order to communicate. They could even be telepathic.
In this case, what would one recognize as grammar? Being able to project a picture into another being's head of exactly what you mean would avoid the need for 'language' altogether.
Although, in order to communicate coherently, any language would need at least some form of consistent structure in order for the aliens to understand each other. So grammar would always be needed in some form or other.
If the language is translated (from shiny lights to English, for example), the grammar would need to be translated too, so I don't think there is much wiggle room to adapt an entirely new system of grammar for a current language.
However, you can always use an incorrect grammar system that is still coherent, as Yoda does. In Terry Prachett's Discworld novels, I remember in at least one that the merchants could not speak with the correct punctuation, but it was still coherently readable. So as long as the grammar is understandable, it could be changed to simply be incorrect.