Assume $n\geq 3$, $p$ is a prime, and that $S_n$ acts on $V=\mathbb F_p^n$ by permuting the basis vectors $v_1,\ldots, v_n$. I want to compute the order of the first cohomology group of this action.
The principal crossed homomorphisms are the functions $f: S_n\to V$ such that there's a $v\in V$ such that $f(\sigma)=\sigma(v)-v$ for all $\sigma\in S_n$. Now, there are clearly at most $p^n$ possible functions, corresponding to choices for $v$. Now, if $\sigma(v)-v=\sigma(w)-w$, then $\sigma(v-w)=(v-w)$. The only way all permutations fix $v-w$ is if $v-w$ were a multiple of $v_1+\cdots+v_n$, so I think that gives $p^{n-1}$ unique principal crossed homomorphisms.
The other thing I have to do is find the number of crossed homomorphisms $f: S_n\to V$, which must satisfy $f(\sigma\tau)=f(\sigma)+\sigma(f(\tau))$. This property immediately gives $f(1)=0$ and $f(\sigma^{-1})=-\sigma^{-1}f(\sigma)$.
If $\sigma$ has order $n$, then we can see that $0=(1+\sigma+\sigma^2+\cdots+\sigma^{n-1})f(\sigma)$. Applying $1+\sigma+\cdots+\sigma^{n-1}$ to a vector gives $\lambda(v_1+\cdots+v_n)$, where $\lambda$ is the sum of the coefficients of $f(\sigma)$, which is thus $0$. This, I seem to have $p^{n-1}$ choices for $f( (1 2\cdots n) )$, though I'm not sure all of them work.
Since $S_n$ is generated by $(1 2\cdots n)$ and $(1 2)$, I can now count the possible images of $(1 2)$. If $\sigma$ has order $2$, then $0=f(\sigma)+\sigma f(\sigma)$, which means that $\sigma$ negates $f(\sigma)$. So if $\sigma=(1 2)$, and $p\neq 2$, then $f(\sigma)$ must be a multiple of $v_1-v_2$, giving me $p$ choices. If $p=2$, then I only need $v_1$ and $v_2$'s coefficients to be the same, so I have $p^{n-1}$ choices.
So this seems to give me $p^n$ crossed homomorphisms if $p$ is odd, and $p^{2n-2}$ if $p$ is even. This would mean that $|H^1(S_n, V)|=p$ for odd $p$ and $p^{n-1}$ for even $p$, but I know that I'm supposed to get cohomology groups of order $1$ and $2$, respectively. Clearly, I'm not completely free to choose the images of the two generators for $S_n$, but I'm not sure how to narrow my choices down. Dummit and Foote also give me a hint to use Shapiro's Lemma, but I have no idea how that applies...
Do you know Shapiro's Lemma? Your module is the induced module of the subgroup $S_{n-1}$ of $S_n$ acting on the trivial module, so by Shapiro's Lemma $H^n(S_n,{\mathbb F}_p^n) \cong H^n(S_{n-1},{\mathbb F}_p)$ for all $n \ge 1$.
For $n=1$, $H^1(S_{n-1},{\mathbb F}_p) = {\rm Hom}(S_{n-1},{\mathbb F}_p)$ which, for $n \ge 3$, has order $2$ when $p=2$ and $1$ otherwise.