  Suppose $q$ is a fixed odd prime power, $F(\vec{x})$ is a non-degenerate
quadratic form over $\mathbb{F}_q[t]$ of discriminant $\Delta$ in $d\geq 5$
variables $\vec{x}$, and $f,g\in\mathbb{F}_q[t]$,
$\boldsymbol{\lambda}\in\mathbb{F}_q[t]^d$. We show that whenever $\text{deg}
f\geq (4+\varepsilon)\text{deg} g+O_{\varepsilon,F}(1)$,
$\gcd(\Delta^{\infty},fg)=O(1)$, and the necessary local conditions are
satisfied, we have a solution $\vec{x}\in\mathbb{F}_q[t]^d$ to $F(\vec{x})=f$
such that $\vec{x}\equiv\boldsymbol{\lambda}\bmod g$. For $d=4$, we show that
the same conclusion holds if we instead have $\text{deg} f\geq
(6+\varepsilon)\text{deg} g+O_{\varepsilon,F}(1)$. This gives us a new proof
(independent of the Ramanujan conjecture over function fields proved by
Drinfeld) that the diameter of any $k$-regular Morgenstern Ramanujan graphs $G$
is at most $(2+\varepsilon)\log_{k-1}|G|+O_{\varepsilon}(1)$. In contrast to
the $d=4$ case, our result is optimal for $d\geq 5$. Our main new contributions
are a stationary phase theorem over function fields for bounding oscillatory
integrals, and a notion of anisotropic cones to circumvent isotropic phenomena
in the function field setting.
