The authors of the work on the amplituhedron (Trnka and Arkani-Hamed) have said that locality and unitarity are "emergent" in this formalism for N=4 Super Yang Mills. The way that locality and unitary appear in these theories is certainly interesting but I think the word "emergent" here is misused.
The usual usage of the word emergence in physics comes from the theory of complex systems where emergent phenomena arise as from the complex interactions of many parts in some limit e.g. on large scales. This means that the emergent system is approximate when viewed at the detailed level of the system. An example is the thermodynamic laws of entropy and temperature which emerge from the statistical physics of molecules.
In N=4 Super Yang Mills locality and unitarity are exact. When you start from the amplituhedron formalism they may not be apparent initially but are derived from the mathematical description. There is no complexity and no approximation, in fact the amplituhedron may be a simpler description that the usual QFT one and it certainly leads to simpler calculations. I think the word "derived" is much more appropriate here than "emergent"
In the question you mention gravitational interactions. Super Yang Mills is not directly a theory of gravity. It is thought to be dual to a theory of gravity due to AdS/CFT but the amplituhedron has not yet cast any light on the duality so that is not the important point here.
Nima Arkani-Hamed likes to compare the situation with the discovery of indeterminsim in quantum mechanics. Before QM was discovered physicists had recast classical mechanics as a principle of least action. In that formalism determinism was no longer directly apparent but it could be derived as an exact principle by solving the equations. Later the principle of least action was deformed by quantum mechanics and then determinism was only an emergent approximation in the classical limit. Here the word emergent is appropriate.
The amplituhedron formalism in this analogy is really just like the principle of least action. Locality and unitarity are still exact but are derived. Nima Arkani-Hamed sometimes makes it sound like they have reached the point analogous to quantum mechanics where locality and unitarity are emergent but that does not yet seem to be the case at least in the work he has talked about.
It is widely thought that locality and possibly even unitarity could be emergent in quantum gravity. Black holes are relevant here but it is not just about large scale black holes. There are arguments that locality fails at small scales because measurement fails when you probe at the Plank scale. The argument is that the energy required becomes so large that small black holes would form and prevent the measurement. Unitarity may fail due to black hole information loss but there is not a consensus on that question. Nima Arkani-Hamed has another argument that unitarity must ultimately fail because all systems are finite.
Super Yang Mills is not a theory of quantum gravity (unless you count AdS/CFT) so there is no reason to expect a breakdown of locality. There has been some progress in applying some of the theory of scattering amplitudes to N=8 supergravity which is a theory of quantum gravity. If that can be related to the amplituhedron we may get a better idea of whether or not locality and/or unitarity are really emergent. If this involved some deformation of the amplituhedron so that locality and/or unitarity are not exact then the analogy with indeterminism in quantum mechanics would be complete. It is too soon to say if things will work out that way or not.