  We consider a stochastic Volterra integral equation with regular
path-dependent coefficients and a Brownian motion as integrator in a
multidimensional setting. Under an imposed absolute continuity condition, the
unique solution is a semimartingale that admits almost surely H\"older
continuous paths. Based on functional It\^o calculus, we prove that the support
of its law in the H\"older norm can be described by a flow of mild solutions to
ordinary integro-differential equations that are constructed by means of the
vertical derivative of the diffusion coefficient.
