did Travis's father go to jail
In attempting to answer your question I see two possibilities for what you are asking:
1. It may be that you are asking why Travis' father was not around, causing Travis to be the one to have to kill Old Yeller
2. It may be that, thinking that Travis' father was the one who actually shot the dog, did he go to jail for killing the animal?
In answer to question number 1, Travis' father's absence was not because he was in jail. The story is set shortly after the Civil War when times were extremely hard and people had to do whatever it took to provide for their families. He and the other men from the area had taken livestock to a distant market to try to make money for their families. Since their only mode of transportation was on horseback or wagon it would take weeks, and sometimes months.
If question number 2 is what is being asked, then the answer is no. Nobody went to jail as a result of killing he dog (though it was actually Travis himself who did the shooting). This was in a time when animals were viewed very differently from the way they are today. Animals actually have "rights" now, and animal cruelty can and often does cause people to go to jail. But such laws were not in existence at that time.
I hope one of these helps.