This is actually a much more difficult issue than it seems on the surface, and much of it has to do with the context that you're using.
There are two contexts for constitutional law: the one that is in use right now, created by years of due process incorporation of the Bill of Rights, and the one that the founders intended, you might say the "originalist" position.
Under the present interpretation of the Constitution that prevails, then of course sharia law which contradicts the Constitution at the state or the local level would likely run afoul of certain Constitutional provisions and protections, and thus would be unconstitutional and impermissible.
But food for thought....
Under the originalist interpretation of the Constitution, the Constitution only applies to the federal government and thus the Bill of Rights only fully applies within federal non-state lands (such as territories and the District of Columbia).
As a result, a community could, under the original Constitutional interpretation, enact sharia law if it so chose, if the state in which it resided allowed it to do so by its laws.
I'm not saying we should allow sharia law in the present context, but it's always worth remembering when we start talking about "is x or y constitutionally permissible" that the original interpretation of our Constitution was that it applied only to the federal government.
Here is a fantastic law review article by one of the greatest living federal judges, J. Harvie Wilkinson III, which talks about the substantive due process overreach present in both Roe v. Wade and DC v. Heller.
Good reading if you're interested in your Constitution...