Quantum cellular automata
Niel de Beausdrap found this article. The authors (Schumacher and Werner) propose a definition of quantum cellular automata and prove they are always invertible (Corrolary 7 on p. 10). Nominally their definition is different since they use the norm-completion of the algebra $A_\Gamma$ and impose a locality condition. However, it is actually equivalent.

Their locality condition states that the image of $A_x$ has to be contained in $A_S$ for some finite S. However, if we avoid the norm completion it holds automatically since any element of the direct limit is in $A_S$ for some finite S and $A_x$ is finite dimensional.

Moreover, an endomorphism of $A_\Gamma$ is always continuous hence extends to the completion. This is because the matrix norm can be defined algebrically as the maximal |lambda| for which a*a - lambda is non-invertible.

Representability follows from their Theorem 6 on p. 10 which gives a rather explicit (modulo some complicated constraints) description of all nearest-neighbour quantum cellular automata. As discussed on p. 9 this description encodes two finite-dimensional unitary operators. These operators can be easily used to construct the operator $U$ I seek.

As discussed on p. 8 the case of non-nearest-neighbour can be easily reduced to this case