Let $M$ be a manifold and $S\subset M$ a submanifold. Assume whatever regularity you need (smoothness, compactenss, orientability, empty boundary, proper embedding of $S$,... this question is motivated by this MSE answer).
Let $N_MS= TM|_S / TS$ denote the normal bundle to $S$. Let $X$ be a normal vector field on $S$, with which I mean a section of $N_MS$ on $S$. Is there a definition of the index of $X$ at one of its zeroes? I only know this concepts for vector fields on a manifold, i.e. sections of the tangent bundle, but here I am interested in the transverse part of the field.