In fact, reality is not apparently basically rational. Why does anything exist at all? Surely, the trivial solution would be a simpler state of affairs. Why is existence not nothing forever? Why there exists anything instead of nothing (let alone you and me, in particular) does not stand to reason.

I imagine those digging would choose to believe they haven’t dug deep enough to understand. It would also be difficult to prove things happen irrationally, and without evidence either way I would expect continued investigation into existence

Most humans believe in magic in some form (gods, spirits, miracles, astrology, etc) so we don't have to guess how people would respond if they believed reality was irrational. They rationalize it endlessly and try desperately not to look at the parts that make no sense.

"There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened." - Douglas Adams

If at some point we reached out and dug deep enough into study of existence only to find that some things simply happen or emerge without any cause whatsoever

I don't think that the human mind would allow that to be a conclusion. Like others have pointed out, some would ascribe the "thing" as happening from a supernatural origin, others would say we just haven't found the reason for it.

I don't think we have evolved to understand that something can happen without any cause.

"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.

Have you seen the world? Shit's irrational as fuck. I'll argue here that rationality has an aspect of subjectiveness to it; It is based on your ability to understand and perceive things. There is a very very small number of things you don't know, and a practically infinite ocean of things you will never even know to not know. Even things that behave rationally may behave irrationally under some variables you will never even fathom, and so, to you, that would just be irrational

You wanna see something rational? Read fiction. Truth is always stranger than fiction. It was Mark Twain who added that "fiction has to make sense."

I mean that sounds pretty close to what absurdism is getting at

It probably would be rationalized up until a loophole in the laws of physics is found, at which point its sheer incomprehensibility would corrupt our existences.

If reality weren't* wholly rational

When using be in an if clause for an unreal conditional sentence, always conjugate it as were, no matter what the subject is. Even if the subject is first-person singular (I) or third-person singular (he, she, or it), still use were with an if clause in unreal conditional sentences.

https://www.grammarly.com/blog/conditional-sentences-was-instead-of-were/

We're not seperate from reality so we fit right in B-)

If at some point we reached out and dug deep enough into study of existence only to find that some things simply happen or emerge without any cause whatsoever...What might be the response?

Isn't this basically a lot of quantum physics? Things pop in and out of existence for no rhyme or reason there. The response is simply "we just don't know enough to understand what we are observing yet." I feel like that sentiment will always be true. If we ever can't rationalize something, we will rationalize why we can't.

Reality already isn't rational in the slightest, and the way humans prevent themselves from going insane is by an innate characteristic of our psychology called cognitive dissonance.