Show that $W = \langle v_1 + \dots + v_n\rangle $ is a submodule of $V$ . Let $G = S_n$. Let $V$ be a vector space over a field $F$ with basis $\{v_1,\dots,v_n\}$. Then $V$ is an $FG$-module with action defined by setting $g · v_i = v_{g(i)}$ for all $g \in G$ and $1\leq i \leq n$ and extending linearly.
i) Show that $W = \langle v_1 +\dots+ v_n\rangle$ is a submodule of $V$.
I can show that for a finite basis that $W$ would be a submodule.
Let $G = S_3$ is the symmetric group on $3$ letters.
Let $V$ be a vector space over a field $F$ with basis $B = \{v_1,v_2,v_3\}$. 
Define an action of $G$ on $V$ by setting $g·v_i =v_{g(i)}$ for $g\in G, i=1,2,3$ and extending linearly. 
So, if $g = (1, 2, 3)$ then
$$g·v_1 =v_2\\ g·v_2 =v_3\\ g·v_3 =v_1,$$ and if $$v=\sum_{i=1}^3 \lambda_iv_i \in V$$ then
$$g \cdot v = \sum_{i=1}^3 \lambda_ig \cdot v_i = \lambda_1v_2 + \lambda_2v_3 + \lambda_3v_1.$$
If $W=\langle v_1+v_2+v_3\rangle$, then $g·w=w $ for all
$g \in G$. So $$W = \langle v_1 + v_2 + v_3\rangle$$ is called a submodule of $V$.
But I am unsure how to apply this to an infinite amount.
Thanks.

There is no "infinite" amount. Your proof for $3$ works just as well for a general $n$.
Take any $g\in S_n$ and any sum
$$v=\lambda \left(\sum_{i=1}^n v_i\right) \in W$$
Now $$g\cdot v=g\cdot \sum_{i=1}^n \lambda v_i = \sum_{i=1}^n \lambda v_{g(i)}=\lambda\sum_{i=1}^n v_{g(i)}=v.$$
