I have this type fixture in my laundry closet and the chain has pulled out/broken. Can if be fixed or will I need to buy a new fixture?

If the chain is broken from the inside of the pull rope switch, it cannot be installed. and we will not suggest fix the broken chain without help of professional. but don'r worry, this pull cord is just an auxiliary switch, this light can also be directly controlled by a wall switch to turn on and off the light. Pay high attention: If the pull cord switch is off, the fixture will not light up even you turn on your wall switch.