Peter Parker is sitting in the car with Adrian Toomes (the Vulture) and they BOTH know who the other one is - so why doesn't Peter simply restrain him and call the police or for help? By this point in the movie he knew where the hideout was already and he could have reported it to the police and say that Adrian showed it to him or something like that.
Please note contrary to your description, Peter didn't know where Vulture's hideout was; he had to leave his cellphone in the car so that Ned could track Vulture back to the building. The only other time he interacted with the villians was when he thought their lair was at the gas station in Maryland, but this was also incorrect intuition, since they were really just parked at the gas station to support the Vulture's heist.
These are common arguments for why Peter didn't just attack the Vulture:
Peter feels that he needs his Spider-Man persona to succeed
The character arc for Peter in this movie is about reconciling his identities as Peter Parker and Spider-Man. After he loses Stark's suit, he appears to give up on being Spider-Man altogether even though he still has his own suit and gear. It is only really seen after he gets trapped under the rubble that he truly recognizes that he is still Spider-Man under any conditions. So it doesn't even enter his mind to try to fight before he has the suit and web fluid.
Peter doesn't know whether he can survive a bullet
Presumably, this Spider-Man has regenerated healing just like the comics. It actually hasn't been established in the movies whether that is the case, or even whether he is less affected by pain (He gets thrown around a lot, but the MCU tends to ignore the affects of a lot of battle wounds for all heroes). So we actually have no way to know whether this Spider-Man would be able to shrug off a bullet. Even if he does retain the comics ability (Which is most likely the case), Peter himself might not know what the capabilities of his regeneration are. We have never seen him get shot before.
Peter doesn't want to fight the Vulture under these conditions
We know that Peter has the confidence to place his phone in the car before leaving, so even though he might have felt intimidated by Vulture, he seems to already have his plan in mind. Maybe he wants to get his web-shooters first because they're his standard way of restraining bad guys. Maybe he doesn't want to fight the Vulture with so many innocent bystanders. Maybe he wants to bust the Vulture at his hideout with his gear so there is no mistaking for police that he really is the Vulture.
Attacking Vulture might have compromised his identity
If Peter successfully overpowers Toomes in the car, his identity might have been guessed after students saw him take down a man with a gun; if he is shot, he could rapidly regenerate from the bullet, which would be suspicious, or he might be forced to go to the hospital after the public fight, where tests might show that he is biologically abnormal. Further, by fighting Toomes in front of so many witnesses, Peter risks Toomes revealing his identity as Spider-Man immediately. There is no good way for Peter to explain why he apprehended the Vulture in the car, since Peter Parker the high-schooler shouldn't have even known about the Vulture at all, let alone be in a situation where he has to fight him because the Vulture is threatening him.
Aunt May told him not to
Peter expresses many times during the movie that he cannot let Aunt May know that he keeps putting himself in danger as Spider-Man. He might feel that he should only take on the Vulture as Spider-Man so that she doesn't later hear that he was fighting with a man with a gun instead of turning around and running away as she instructed him to do in dangerous situations.
Some of these are more unlikely than others, but I'm sure there's at least one of them that sounds good for you. I've always felt that the first one was the strongest argument, since we see Peter Parker as the scared kid in the car who then runs to change into his Spider-Man persona so he can take control of the situation.