Vue JS mounted()

Abstract your initialization into a method, and call the method from mounted and wherever else you want.

new Vue({
  methods:{
    init(){
      //call API
      //Setup game
    }
  },
  mounted(){
    this.init()
  }
})

Then possibly have a button in your template to start over.

<button v-if="playerWon" @click="init">Play Again</button>

In this button, playerWon represents a boolean value in your data that you would set when the player wins the game so the button appears. You would set it back to false in init.