The Night Monkey suit only appeared when he was fighting the Fire Elemental monster.
Up to this point, all actions of the drones were carefully crafted illusions, holographic projections and pyrotechnics programmed beforehand by Mysterio and his team.
It is only after Mysterio obtains the E.D.I.T.H. sunglasses that they have broader control of actual firepower from both satellite and drone artillery.
Indeed, it is when Spider-Man discovers Mysterio's secret plans do they dial up the lethal weapons to 11.
Consequently, in my opinion, the stealth suit (the proper name) is not made of any special material other than what was available to S.H.I.E.L.D. not necessarily bulletproof Stark tech