For companies to cause problems they have to buy so many homes they can abuse their market share by forcing rents up
Not necessarily. Your logic only applies to non-necessities. For a necessity, all you need is to own enough of the industry, that you're the only option for some people. If there only exists enough housing to just barely house everyone who needs a home, then you could own only 1%, and set the price to whatever you want, because someone will have to live there. Everywhere else is occupied.
which you would see as an increasing vacancy rate
No, you wouldn't. There is no rent that is so high people will decide to live on the street. They'll keep paying, until they actually can't.
Again, this logic only applies to non-necessities. If you hike the price of food, people don't stop eating. They stop bying luxury goods.
And no, if there isn't an overabundance of food (as you yourself admit, housing is insufficient) they can't just switch to the competition, because the competition does not have the capacity to serve everyone.