First, just want to clarify some terminology, which I know can be a bit confusing when new to the site. Less Wrong is about rationality rather than rationalism, which are linguistically close and not totally unrelated, but are used as jargon to mean different things. The short way to understand it is that rationality is about having accurate beliefs, rationalism is the philosophical stance that reason is the primary source of true knowledge, as opposed to observation, prior beliefs, etc.

Second, I wrote a series of posts about the intersection of rationality and zen, since I practice both. Maybe I'll get back to adding to it one day. You might find that interesting.

Third, let's address your question, on the assumption that you meant "rationality" by "rationalism" (sorry if this is a wrong assumption!). So, to the extent that the theory of dependent origination is correct, then rational agents should place proportional credence in its truth. That said, dependent origination is a metaphysical claim, thus hard to test, so most rational agents will be forced to be relatively uncertain about its truth because we are limited in our ability to check its truth. For example, from inside the world, we can't distinguish between one where dependent origination is true and one where the world exists for exactly the single moment when you bothered to check and it just happened to be arranged in a way that made it looked like there was cause and effect, though the former belief is also obviously more useful than the latter even if you can't tell which one is really the true nature of the world.

On the other side, rationality is deeply tied up with the theory of causation because an understanding of causation is necessary to make sense of much of the evidence we observe and make updates to our beliefs that are going to be as accurate as possible. I think most rationalists implicitly believe in something like dependent origination even if they don't have an explicit theory of it because it's (mostly) part of the standard Western model of the world. Modern Westerners generally seem to understand that everything is causally connected within a given thing's Hubble volume, even if some of those connections are quite weak. So all this is to say that rationality's compatibility with dependent origination is not intentional, but because dependent origination, under different names, is now just the background assumption of the type of people who become rationalists.

That said, two additional thoughts. One, just because it's the background assumption doesn't mean people understand it well if you press them for details, and you can easily get people to claim that things are not causally connected because they are unreflectively ignoring causes that don't fit within their ontology. Second, the teaching on dependent origination is best understood in context, where it was given at a time when many people had a model of the world that suggested many things had independent causes, or maybe even no causes at all! So it's worth looking at the connection, but also don't be surprised if you dig into it and find everything adding up to normality.

I think you could apply rationality even in a universe with a random number generator, as long as most things are causal.

Even if the universe is causal, we still need a good strategy to think about it (i.e. rationality).