"Rationality" is perhaps a more flexible concept than what you're assuming. As long as there's patterns to reality then there will be ways to come up with rules to describe them, and make predictions about future events.
Causality, in particular, is not strictly necessary for rationality. Our understanding of the laws of physics already accounts for situations with uncaused events, in fact. Radioactive decay is an example; an unstable atomic nucleus sitting alone in space with no outside interactions will sometimes spontaneously decay with no preceding "cause". Virtual particles are another - subatomic particles can spontaneously pop into existence and then pop back out again without a specific event causing it.
There are also proposed laws of physics for handling the concept of time travel that can allow for causal loops - things from the future affecting their past selves. We have no reason to believe that this is actually possible, but if one day we discovered that it was possible then it can still be accounted for using rational means.
So I'm not really sure it's possible to have a universe that isn't wholly rational. Even pure chaos is still describable and predictable in its very unpredictability.