The wall was not built to keep wildlings away as it has been a common fact in the world of Westeros before all of these NK related thing happened. So why was it built? NK seems powerful enough to gather an army of the dead and employ them to storm the giant gates and break open them to pass through it, he didn't need a Dragon to first tear down the Wall. However that did make it easy for his army to pass the wall.
But, there's a catch, Wall isn't (well, wasn't actually) just a physical barrier but also a magical spell was cast on the Wall to keep the Night King North of the Wall, the army of dead however were allowed through the wall but not the NK or his White Walker team. We know that because we have seen some Undead coming south of the wall, like in the first season when the magic spell was still active on the Wall.
Then, the cave in which three-eyed raven was also protected by not any physical barrier but only and only magic, so he was never able to go past it.
However, when NK left a mark on Bran the magic spell broke and he was able to go pass the boundaries of that Weirwood tree where three-eyed raven resided.
So coming back to your question, even if he had no dragons it would have been possible for him to pass through the wall given that Bran had his mark.