How is the flow of air through a NACA submerged inlet determined?
The flow through the inlet is determined by the resistance of everything behind the inlet. If the inlet is blocked off internally, there is no flow and $v_i$ = zero.

The mass flow is indeed the way to determine through-flow: mass in = mass out. Which is a function of the pressure build-up (both in and out), cross section area (both in and out), temperatures, etc. A similar process occurs in a jet engine inlet.