That depends on how long this goes on. It takes months to build a new unit, so it will make no difference in 3 months. In many areas there is a lot of pent up demand from people with more means who are living elsewhere, and they are now living in houses in areas far enough away that those living in tents won't find moving there useful.
In short it will take a decade or two of building to make a difference to those living in tents.
Also living in tents often implies other issues. More housing will not solve anything if your real issue is serious mental illness. There will also be a few people living in tents/on the street because their reason for living in tents is not related to the cost of housing. Building more housing will make a difference to those who's problem is housing is too expensive, but it can't help those who's problems are not high cost of housing.