How far from their home do lynx go to eat?

It depends on how much food is available (the scarcer the food the further the Lynx will need to travel).

When thinking about how far a lynx needs to go to eat it is normal to think about the area of an individual lynx's territory (lynx will have neighbors all around) and lynx territory sizes ranges from about 25 to 2000 sq km according to habitat quality and prey availability