One thing to know right away is that localStorage can only save strings. Therefore when you want to save a non string value, you need to call JSON.stringify(data) before saving, and JSON.parse(data) after getting the data and before using it.

In your case, an easy way is to add the player and then save prevRanking as string. But before you should parse what you get as it is always a string if not null.

  function handleRanking() {
    console.log('run function');
    let prevRanking = localStorage.getItem('ranking');
    if (prevRanking === null) {
      console.log('null value');
      prevRanking = [];
    }else{
     prevRanking = JSON.parse(prevRanking);
    }
    const player = {
      name,
      score: total,
      picture: gravatarEmail,
    };
    if (Array.isArray(prevRanking)) {
      prevRanking.push(player);
      localStorage.setItem('ranking', JSON.stringify(prevRanking));
    } else {
      console.log('not an array');
    }
  }

After you have retrieved the info from the localStorage you need to pass it into the JSON.parse method. Because when storing information in the localstorage it is saved as string.

function handleRanking() {
    console.log('run function');
    let prevRanking = localStorage.getItem('ranking');
    if (prevRanking === null) {
      console.log('null value');
      prevRanking = [];
    }
    prevRanking = JSON.parse(prevRanking); // added it in here so prevRanking is definetly defined
    const player = {
      name,
      score: total,
      picture: gravatarEmail,
    };
    if (Array.isArray(prevRanking)) {
      prevRanking.push(JSON.stringify(player));
      localStorage.setItem('ranking', JSON.stringify(prevRanking));
    } else {
      console.log('not an array');
    }
  }

Because the value of a key gets from LocalStorage is a string. So, you have to JSON.parse before pushing. https://developer.mozilla.org/en-US/docs/Web/API/Storage/setItem#parameters