Probably not. Most homelessness is due to either mental illness and substance abuse or due to a shortage of affordable housing in places where jobs are available - typically dense cities where the only way to add housing within reasonable commute distance of the jobs is to build upwards, not building inexpensive 1-story buildings on vacant land.

Furthermore, both types of homeless tend to congregate in places where there's a density of social services available because they rely on those services for food, clothing, health care, or assistance seeking employment, in addition to shelter. Those areas tend not to be within walking distance (for the vast majority who also lack cars) of anywhere that has space to build dozens or hundreds of these tiny houses.

There are plenty of places in this country that have a ton of vacant housing stock, and even some of those have a homelessness problem (see also: Detroit). The problems underlying homelessness are far more complicated than the size of the houses we build.