The following results easily follows from inverse function theorem:
Theorem. Let $U$ be an open set in $\mathbb{R}^n$, $p\in U$ and $X:U\rightarrow\mathbb{R}^n$ a $\mathcal{C}^k(U,\mathbb{R}^n)$-vector-field with $k\geqslant 1$. If $X(p)$ is nonzero, there exists $V$ an open neighborhood of $0$ in $\mathbb{R}^n$, $W$ an open neighborhood of $p$ in $U$ and $f:V\rightarrow W$ a $\mathcal{C}^k$-diffeomorphism such that $f(0)=p$ and $f^*X\equiv X(p)$.
I am strongly convinced that from there one can establish the following:
Theorem. A $\mathcal{C}^1$-vector-field on $\mathbb{S}^2$ vanishes at at least one point.
Assume there exists $v$ a $\mathcal{C}^1$-vector-field on $\mathbb{S}^2$ that does not vanish on $\mathbb{S}^2$. Let $p_N:\mathbb{S}^2\setminus\{N\}\rightarrow\mathbb{R}^2$ be the stereographic projection of north pole onto the equatorial plane. Then, $w_N:=p_N\circ v$ is a $\mathcal{C}^1$-vector field of $\mathbb{R}^2$ that does not vanish. Hence, there exists a $\mathcal{C}^1$-diffeomorphism from an open neighborhood of $0$ onto an open neighborhood of $S$ such that $f^*v$ is constant. I cannot proceed any further from here.
My goal is to deduce hairy ball theorem from the inverse function theorem. Thus, I would like to avoid arguments using the fundamental group of $\mathbb{S}^1$ and degree theory.
Any enlightenment would be greatly appreciated!