Given that Shor's algorithm is mostly Toffolis, it would be nice to run these gates transversally while leaving the less used gates up to distillation/cultivation. There are codes with transversal CCZs but this would call for the need to teleport Hadamards most of the time in order to execute the Toffolis, which is not ideal.
What is a quantum error correcting code with a transversal Toffoli?
Not really, at least not in the way that you want. Quantum polynomial codes admit a "kind-of" transversal Toffoli, but really what happens is that it transforms between codes that eventually run out of steam (and I think run out of steam "exponentially quickly"... really what it does is increase the degree of the polynomial encoding the target, which you can't do for long). See e.g. section 8.3.4 of https://dspace.mit.edu/handle/1721.1/44407?show=full.
In the stricter sense, for stabilizer codes, you can show that all transversal gate groups belong to the Clifford hierarchy. Since Toffoli generates gates outside of the Clifford hierarchy, it cannot be transversal. See e.g. https://arxiv.org/abs/1710.07256.
