What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)

In set theory, $A^B$ usually means the set of all functions from $B$ to $A$. In that sense, you can see $A^n$ as the set of all functions from $n$ to $A$ where $n$ is the von Neumann ordinal $\{0,1,2,\dots,n-1\}$ (which is a set with $n$ elements). In that sense, an $n$-tuple of elements of $A$ and a function from $n$ to $A$ are just two different ways of interpreting the same thing.

So, $A^0$ would then just be the set of all functions from the von Neumann ordinal $0$ (which is the empty set $\emptyset$) to $A$. And there's only one such function which is $\emptyset$, so $A^0$ must be $\{\emptyset\}$ - a set with one element.

This all fits perfectly. It seems you have no issues with, say, $2^0$ being defined as $1$. This is a similar construction. Actually, $A^0=A^\emptyset=\{\emptyset\}$ not only has cardinality $1$ but is $1$ in the von Neumann sense.

(And this interpretation makes a $0$-ary function a constant, BTW.)