Hcfyk

Do hobos exist in The Hunger Games? Or does everyone have a house or shelter to live in?

We learn in Catching Fire that the people in District 12 have "designated dwellings", so it seems doubtful that there are any homeless people:

My old home. We still get to keep it since officially it's the
designated dwelling of my mother and sister. If I should drop dead right now, they would have to return to it.

The point of this obviously is to control the people, not to be generous.

When people flee from their District, they live outside. What happens from there is usually that they end up dead, killed, captured and turned into Avoxes...except for the lucky ones that end up in District 13, which takes in all refugees. In fact, in Mockingjay it takes in all 800+ District 12 refugees who had been wandering the woods for three days after the District was destroyed.

As for the Capitol, it's even less likely that they have homeless people, at least under normal circumstances. The Capitol is rich and has the sort of people who just wouldn't be able to live on the streets. Plus, in Mockingjay, when the rebels attack the Capitol by using cars to detonate pods (Chapter 24), we see refugees being forced into homes:

Many are still out on the street, trying to find shelter for the night. Those who live in the choice apartments of the inner city have not flung open their doors to house the displaced. On the contrary, most of them bolted their locks, drew their shutters, and pretended to be out. Now the City Circle's packed with refugees, and the Peacekeepers are going door to door, breaking into places if they have to, to assign houseguests.

On the television, we watch a terse Head Peacekeeper lay out specific rules regarding how many people per square foot each resident will be expected to take in. He reminds the citizens of the Capitol that temperatures will drop well below freezing tonight and warns them that their president expects them to be not only willing but enthusiastic hosts in this time of crisis. Then they show some very staged-looking shots of concerned citizens welcoming grateful refugees into their homes. The Head Peacekeeper says the president himself has ordered part of his mansion readied to receive citizens tomorrow. He adds that shopkeepers should also be prepared to lend their floor space if requested.

This would work well if there were only a few people who needed homes, but there were too many. People spend the night in the street and are tightly packed into stores.