What is so important about taking Calculus III before Linear Algebra? Is Calculus II sufficient?

If memory serves me I think I took Calc III and Linear Algebra simultaneously without any difficulty.  Calc I should teach you derivatives and Calc II integrals.  That should really be more than sufficient to understand Linear Algebra which really shouldn't involve much, if any, calculus at all.

If anything you would want Linear Algebra as a prerequisite for Calculus III, not the other way around.  Linear Algebra would teach you some determinant theorems which would be useful for working with Jacobians in Calc III.

Have you spoken to your advisor and professors about these prereqs?  Are you sure they didn't advise you to take Calculus III 'INSTEAD OF' (not 'before') Linear Algebra?  Certainly I'd imagine that learning more calculus would help you far more with statistics than matrices would (but I could be wrong).