Where rents are high supply is less than demand - as demand is reduced by people living away and by people cohabiting when they would rather not.
On the opposite situation where properties are empty, they are empty at any price, LVT won’t change the price. They currently can’t be rented out for $50 a month, so why would they be rentable with LVT
The only properties which will be flakey are those where the cost of renting it out (maintenance etc) is lower than the current asking price, but higher once you include LVT.
However as LVT will cost whether you’re renting out or not this will simply increase incentive to sell (if you can’t get enough to rent out), reducing the purchase price, and allowing people to live there.
Given that in cities where rents and land values are highest the number of empty properties is minimal, increasing the cost of owning an empty property won’t increase that number.