Electric Field at center of hollow sphere split into two hemispheres

I think some more clarification is needed. Is the sphere perfectly conducting? Is it a conducting shell with vacuum inside? Are you assuming a charge distribution on the shell that creates the electric field? It is definitely possible to have a non-zero field in the center of a sphere if the charge distribution is asymmetric.

I hope I've understood your question right. In case of two perfectly conducting hemispheres separated by an infinitesimal layer of vacuum that prevents the charges from moving, the charges on the spheres would not remain uniformly distributed, but redistribute themselves on the surfaces of the spheres. They will redistribute in such a way that the electric field inside the conductors vanishes everywhere, in a manner somewhat similar to that shown here:

Between the two hemispheres (neglecting fringing effects at the edges), the field will look like that of a parallel plate capacitor of infinitesimal thickness. In the layer of vacuum between the hemispheres, there will be an electric field pointing from the positive to the negative charges and of magnitude $E = $rho_s / epsilon_0$, where $rho_s$ is the (positive) surface charge density on the charged sphere in this region. There is no problem having an electric field in the center of the sphere in this case.