Vacancy rate statistics include things that are pretty natural and you wouldn't really want to penalize, like an apartment being briefly vacant after the previous tenant moves out and before the next tenant moves in. So you can't just look at the rental vacancy rate in a city being (say) 3% and conclude that those 3% of units in the city are being held vacant and off-market. There may be some of those, but it's a far lower percentage.
Which is to say, institute a vacancy tax if you like (it won't really hurt if done sanely), but realize that it won't be a silver bullet here.
In California, for example, you have reports of people who move and use their old home in the Bay Area for storage because they're paying the same nominal property tax rate from 40 years ago. If they were paying a yearly 1% tax on the value of their property (instead of <0.05%), they'd rent or sell it immediately.
Same deal in China, except more extreme. Property _is not taxed_, so there are/were a lot of people building apartments and leaving them vacant, waiting for prices to rise over time.
Neither of these situations serve the material needs of people today who need cheap, high-quality, and abundant shelter.
Plus, your vacant property tax proposal excludes the (500k+?) many empty bedrooms in occupied homes. If someone's kids have grown and moved out, their old rooms are now vacant. How many people simply leave these rooms vacant for years and decades? A blanket property tax increase would strongly incentivize the renting of these rooms.
This is a myth. Nobody is paying the same property tax from 40 years ago. Property taxes in California go up 2% every year under Prop 13.
It can be argued whether costs should increase faster or not, but in any case it is factually incorrect to say someone is paying the same as 40 years ago.
This would place more downward pressure on rents and decrease the investment upside on buying.
I'm sure this is a bad idea in a number of ways I haven't considered.