This line of thought seems to be the essence of the research program of Gerard 't Hooft, as exposed in a series of papers culminating in the monograph The Cellular Automaton Interpretation of Quantum Mechanics.
A cellular automaton is a system with localised, classical, discrete degrees of freedom, typically arranged in a lattice, which obey evolution equations. The evolution law for the data in every cell only depends on the data in the adjacent cells, and not on what happens at larger distances. This is a desirable form of locality, which indeed ensures that information cannot spread faster than the speed of light. The Cellular Automaton Theory assumes that, once a universal Schrödinger equation has been identified that encapsulates all conceivable phenomena in the universe (a Grand Unified Theory, or a Theory for Everything), it will feature an ontological basis that maps the system into a classical automaton.
The Cellular Automaton Interpretation of quantum mechanics suggests to us what it is that we actually do when we solve a Schrödinger equation. We thought that we are following an infinite set of different worlds, each with some given amplitude, and the final events that we deduce from our calculations depend on what happens in all these worlds. This is an illusion. There is no infinity of different worlds, there is just one, but we are using the “wrong” basis to describe it, because the basis we are using is not an ontological one.
So, at least according to 't Hooft, the answer to the question of the OP "Is it possible for [a local cellular automaton] to be consistent with current physical observations?" is Yes.