You can have $m=n+1$. Take the vertices of a regular simplex
with centre at the origin.

You can't have $m=n+2$. There is at least a two-dimensional space
of vectors $(a_1,\ldots,a_{n+2})$ such that
$$\sum_{i=1}^{n+2} a_i v_i=0.$$
This gives enough room for manoeuvre
to ensure some $a_i>0$ and some $a_j<0$. Thus we get some
nontrivial relation
$$\sum_{i\in I}a_i v_i=\sum_{j\in J}b_j v_j\qquad\qquad(*)$$
where all the $a_i>0$ and $b_j>0$ and $I$ and $J$ are disjoint non-empty
sets of indices. It follows that the dot product of the two sides
of $(*)$ is negative, but that contradicts it being the square of
the length of
the left side.