Is it possible to have a state of the universe with no solution to what state it will evolve into in general relativity?

Your question appears to be unanswerable being based on a misinterpretation of what "closed time-like curve" means. Wikipedia states :"The photon sphere is a spherical boundary of zero thickness in which photons that move on tangents to that sphere would be trapped in a circular orbit about the black hole." A circular orbit is not necessarily a closed curve though. It would be a closed curve, if it ends up where it starts. That this isn't the case as is shown here photon sphere Fig.3.469.

Closed time-like and light-like curves (which are not geodesics) exist in Gödel's universe.

Let me add that if you drop "closed" then your question "I'm wondering if in general relativity, it's possible to start with a state of the universe that will evolve into a state with a (closed) light-like curve with no solution to the state of the universe anywhere in the future light cone of that closed light-like." would refer to the circular orbit of photons (which are light-like curves) in the photon sphere of a static black hole. The evolution of an astrophysical black hole, the collapse of a spherical symmetric cloud of matter, was described by Oppenheimer & Snyder. But once the event horizon is formed the future light cone at $r=3M$ (photon sphere) is tilted toward the black hole with no further evolution because the Schwarzschild spacetime is static outside the event horizon.