Some starting numbers here would be that the leading hypersonic projectile will be scooping up a cylinder of atmosphere equal to the frontal area and it's traveled path. The first approximation of the distance that will halve the projectile speed will be the point at which the mass of that block of atmosphere equals the mass of the projectile.
Mass of atmosphere over a point at sea level 1kg per square, so clearing 100km of air going straight up with a 10m by 10m projectile involves shifting something like 1000 tonnes of air, 1.4 times that if shooting at a 45 degree angle, so your 10m by 10m projectile is weighing that sort of amount as well to achieve useful clearance.
Once you've fired your 1000 tonne bullet the air will try to return, and the gases move at their speed of sound, so 300 m/s for sea level air. This means that the 10 meter tunnel will be well on the way to closing in 0.015 of a second, though pressure and inertia will be at work here as well slowing the return, and producing some interesting pressure effects as the shock front meets. The flight path up the vacuum channel is probably unstable, with a potential for drag forces to amplify any asymmetry and try to tumble the projectile out of the channel.
So in physical terms it can work to make a vacuum but you need to fire very heavy projectiles fractions of a second apart (possibly with active guidence). If going down this path the answer is very high sustained rates of fire. If you can shoot 1k tonne projectiles every 0.05 of a second then your projectile stream will keep a low pressure zone in place, and probably a decent high speed tail wind.
Though power demands of doing this would be substantial. twenty 1k tonne projectiles a second to 10kms is 50 T watts, about 50 times more the total earth capacity. Actual launch velocity probably needs to be higher, depending on how much of that 1000 tonnes is second stage rocket to stabilise the final orbit.
Note also that bringing this system online will produce a rain of 1000 tonne projectiles at various locations down range as the chanel builds. Any misfeeds in the 20 rounds a second feed system will also require rebuilding the channel, prducing more short landings.
So while the physics can be abused enough to work it does not make a very practical space launch system by itself and would be very dangerous to be around in many many way.