More of a comment than an answer, but it didn't fit:
In fact, the hairy ball theorem works on any even-dimensional sphere. However, odd-dimensional spheres do admit nonvanishing vector fields: if you use the usual embedding of $S^{2k-1}$ in $\mathbb{R}^{2k}$, then at the point $(x_1,y_1,\ldots,x_k,y_k)$ you can put the (nonzero) tangent vector $(y_1,-x_1,\ldots,y_k,-x_k)$.
The natural generalization is: How many linearly independent tangent vector fields can exist on $S^{n-1}$? As it turns out, the answer only depends on how many factors of 2 there are in $n$. So e.g. $S^3$ and $S^{35}$ both admit exactly 3 linearly independent tangent vector fields, because $3+1=2^2$ and $35+1=2^2\cdot 9$. This is crazy!