Could we use permutation polynomials for Shamir's secret sharing scheme?

Short answer: No

A permutation polynomial is a polynomial $f:\mathbb{Z}_p\rightarrow\mathbb{Z}_p$ which is a bijection meaning the list $[f(x): x \in \mathbb{Z}_p]$ is a permutation of the elements of the field.

Example: For example $f(x)=x^3$ gives the list $[0,1,3,2,4]$ as $x$ ranges over $\mathbb{Z}_5$.

But these polynomials leak information since if you know $(x_0,f(x_0))$ you know for all $x\neq x_0$ the value of $f$ is different than $f(x_0)$.

This means that the Shamir argument about the polynomial values being uniformly distributed if some $s$ shares for $<s<t$ are known no longer holds. Here $t$ is the threshold.

So, no need to do this it introduces a weakness.