Category theory uses morphisms, which are associative by definition. This means that non-associative operations cannot be trivially mapped into morphisms. One must find clever ways to deal with this. For instance, loops and quasigroups have an operation that cannot be treated as a morphism because it is non-associative. Instead, one must construct a system which is isomorphic to these loops with twice as many operators: "multiply on the left" operators and "multiply on the right" operators (The $L_x$ and $R_x$ operators on the Wikipedia page linked above)
So, in this case, we can study loops because we can construct an associative operation which is isomorphic to the loop operator. But can this be done in general? Can we always fully study a non-associative system in this way? Or are there systems which defy such manipulation, such that some statements about the system cannot be made using category theory but can be made using a proof system tailored to the non-associative system?