If we use Euclid's representation for integers $n=aq+r$, we can write $n\equiv r \mod q$.
We can also write functions similarly, for example $n(x)=a(x)q(x)+r(x)$ and so I imagine we can write $n(x)\equiv r(x)\mod q(x)$.
There are loads of theorems using the arithmetic modular, for example Wilson, Euler and Fermat. Are there any known theorems for the functional modular?
(I am thinking of the associate $b$ of a given $a$ such that $ab\equiv 1\mod p$)
Assuming your "functions" are polynomials over the integers, you're talking about the commutative rings $\mathbb Z[x]/\langle q[x]\rangle$. Yes, a lot is known about these.