My question is whether designing a non-periodic tiling is computable.
The answer is yes, absolutely. The 1966 construction of Robert Berger produces an infinite family of aperiodic tile-sets. (That is, each of the tile sets it constructs will tile the plane, but none of them will tile it periodically.)
The method goes like this:
Pick a recursive language and construct a Turing machine that decides it. This machine will halt on all possible inputs.
Use Berger's construction to translate the machine into an aperiodic tile set.
Each infinite row of tiles represents one state of the machine's execution, including its infinite tape, its current head position and internal state.
The row with $y=0$ represents the initial execution state, and the tiles can be constructed so that this row can only be extended to infinity in ways that actually represent a real execution state. For example, a subset of the tiles will be special tiles that mean “the tape head is at this location”, and the tile set can be constructed so that every infinite row of tiles must include exactly one of these special tiles.
The tiles are constructed so that an adjacent row of tiles must then represent the next complete execution state. This extends to a covering of the upper half-plane. It is periodic only if the execution falls into an infinite loop. But since, by construction, the original Turing machine is known to halt on all inputs, this is impossible.
(The compactness theorem can be used to show that any tiling that covers the half-plane can be extended to cover the whole plane, so this detail can be ignored.)
The tiles are usually expressed as colored squares, where adjacent colors must match; these are called “Wang tiles”. But it is easy to convert Wang tiles into uncolored squares with wiggly edges that fit together like jigsaw puzzle pieces, making the constructed tiles purely geometric.
We can turn the Berger construction around: given a tile-set, can we decide whether it will tile the plane periodically? No, because by Rice's theorem we know that it is undecidable whether a given Turing machine goes into an infinite loop on every possible input.
Of course this does nothing to help decide whether a particular set of shapes has a periodic tiling, or an aperiodic tiling, or even any tiling at all. But as you said, your question was whether designing a non-periodic tiling is computable, and the answer to this question is definitively yes.