Vacant doesn't mean available. Apartments that are being shown to renters are vacant. Houses for sale are vacant. Houses that have sold or rented but haven't been moved into are not available.
It would be impossible to house homeless in vacant houses. Sellers and landlords would never accept it because would make it hard to find people. The short-term rentals would be hard to mange, not good for the homeless to always be moving, and easily go wrong. Is the government going to compensate for damages? For delayed moving into new house? It is almost certainly cheaper to build the homeless housing.
It would be impossible to house homeless in vacant houses. Sellers and landlords would never accept it because would make it hard to find people. The short-term rentals would be hard to mange, not good for the homeless to always be moving, and easily go wrong. Is the government going to compensate for damages? For delayed moving into new house? It is almost certainly cheaper to build the homeless housing.