Well, there's no problem if the map $f$ is real-analytic and close enough to the identity. Then it essentially reduces to the Cauchy-Kovalevskaya Theorem. For the smooth theory, you'll need something a bit more subtle, but it may be OK anyway. The characteristic variety (again, assuming that the initial conditions are sufficiently close to the identity) factors into three linear factors, which is something that Dennis DeTurck and Deane Yang know quite a bit about. It's possible that this problem is what they call `symmetric`symmetric hyperbolic', in which case, the initial value problem will be OK. I'd check with one of them.
Oh: I should have explained that, because the characteristic variety is not smooth (being the union of three lines in general position in the projectivized cotangent space of each point of the domain, the initial value problem is not symmetric hyperbolic in the strict sense, so the simple hyperbolic existence theory for smooth initial data cannot be applied. That doesn't mean (as mentioned above) that a refinement of the symmetric hyperbolic theory won't work. The difference between this and the 2D case is that, in that case, the characteristic variety is 2 (real distinct, multiplicity one) points in projectivized cotangent space of each domain point, so, of course, it is smooth, and the standard hyperbolic existence theory applies.
By the way, you don't have uniqueness. When the obvious inequalities on the initial data are satisfied (so that the hyperbolic theory can be applied), there will be two (and only two) distinct solutions for each choice of initial data $f$ that you prescribe along $\mathbb{W}$.