So, this isn't my wheelhouse, but I was curious too. It seems from the research that Reagan's privatization of the mental healthcare community contributed somewhat, but to credit it to Reagan himself is specious at best:

"Chief among these not-very-important factors is the ongoing movement to deinstitutionalize the mentally ill. I do not mean to make light of the very serious mental health problems faced by many homeless people, but it is simply wrong to suggest that most, or even many, people are homeless because they have recently been released from a mental hospital. What is sometimes overlooked in discussions of homelessness and deinstitutionalization is that we began deinstitutionalizing the mentally ill in the 1950s. The movement accelerated in the 1960s, owing largely to some favorable court orders concerning less-restrictive treatments. By the late 1970s, most of the people destined ever to be "deinstitutionalized" already had been. So as a direct contributing factor to the rise of homelessness in the 1980s, deinstitutionalization cannot be that important. Many also seem to think that the rise of homelessness in the 1980s was the direct responsibility of Ronald Reagan particularly the result of the many cutbacks that Reagan engineered in human-services spending. This, like the deinstitutionalization theme, is at best a half-truth. Certainly, the Reagan years were not kind or gentle to the nation's poor, destitute, and homeless. . .” -- James D. Wright, Address Unknown: Homelessness in Contemporary America, Society, September/October 1989. p. 52.

However, Gittelman seems to suggest otherwise - yet he makes one passing reference rather glibly in a paragraph before going on to his larger issue- and doesn't address the larger trend of deinstitutionalization described (all the way back in 1989, no less!) by Wright:

"Homelessness has grown remarkably since the 1980s, particularly in the United States with 2.75 million persons disabled by mental illness, when the state began deinstitutionalizing the mentally ill and shifting responsibility for their care away from the staff of hospitals, they were suddenly “freed” as the public saw it, which meant, in effect, that for many they were free to live under a bridge and forage in the trash, if they so wished. The decade also saw a shift in reduction in social security disabled benefits shift to state and local from federal less noticed, responsibility for supplying basic necessities (e.g., food, clothing, etc.) away from government to religious and other nonprofit charitable organizations . . .” -- Martin Gittelman, the Neglected Disaster: Homeless and Mental Illness, International Journal of Mental Health, vol. 43., no.2, Summer 2005. pp. 9-21.

Here's a huge point you need to remember whenever discussing increases or decreases in the homeless population: statistics regarding homelessness are necessarily estimates in many cases. To make a definitive statement about the increase or decrease in homelessness is one thing, and hard. To attribute that growth or decay to one President's public policy is entirely another, and harder.