A policy like this would apply to the entire market. All landlords have vacancies at least occasionally, due to renovations or bad luck.

Wouldn't it only apply to the local market? A lot of people, particularly higher earning white collar workers have the ability to demand a work from home policy. Could they not move further away to cooler markets if their commute is eliminated or reduced to only a few days per week? Surely that would put downwards pressure on the inflated local market, moreso if a progressive tax system is implemented (eg tax rates increase % after value increases by a certain threshold).

It won't affect a tenant's ability to pay more, but a policy that increases ownership costs across the board means that there won't be cheaper alternatives in the competition, so the tenant will need to either find a way to pay the increase or they'll have to leave to a cheaper market. The highest rent the market can bear will go up if it's not possible to compete any further on lower prices.

Unless I'm grossly misunderstanding how land tax works, it won't evenly apply across the board (even a flat % tax would be a higher burden for more expensive properties). This would drive people towards constructing cheaper housing as the bottom falls out of the top end of the market, which in turn would make housing cheaper for owner occupiers in those cheaper markets. Isn't that the desired outcome?