In ordinary category theory there is a notion of a diagram in a category $\mathsf{C}$ which is usually described as a functor $F: \mathsf{J \to C}$ where $\mathsf{J}$ is some small category. Based on this we can talk about existence of limits/colimits of $F$, and other properties etc.
Is there a notion of diagram that generalizes to multicategories or operads? I'm reading Leinster's Higher Operads, Higher Categories and it's unclear to me if such a notion exists. I'm trying to assess this for a research question in which I'm trying to describe something as an operad, but this only makes sense for my purposes if I can investigate diagrams and the notion of colimit.
Yes, of course. A diagram is just a fancy name for a functor. A functor is a morphism of multi-categories/colored operads concentrated in arity 1 (AKA categories). The obvious generalization is morphism of multi-categories/colored operads. You send objects to objects (colors to colors), and you send arrows to arrows, respecting target/source colors, composition, and units.
As for limits/colimits, I don't quite see how one would make sense of that. In my opinion, names should be thought of after coming up with definitions, not before.
As someone mentioned in the comments, Lurie has defined something called "weak operadic q-colimit diagram". Maybe it's what you're after, maybe not. But this seems to be quite specific (the diagrams considered are only inductive limits over $\mathbb{N}$, and the main property sought is commutation with the tensor product).