This line of thought seems to be the essence of the research program of <A HREF="https://en.wikipedia.org/wiki/Gerard_%27t_Hooft">Gerard 't Hooft</A>, as exposed in a series of papers culminating in the monograph <A HREF="http://arxiv.org/abs/1405.1548">The Cellular Automaton Interpretation of Quantum Mechanics.</A>

> 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**.