JSFiddle - React, Tailwind, and code Playground

HTML

<p id="json_clean"></p>

JavaScript

data = [
    {
        "State": "California",
        "Cities": [
            {
                "City_Name": "Long Beach",
                "Museum_Name": "Museum of Latin American Art",
                "link": "link"
            },
            {
                "City_Name": "Los Angeles",
                "Museum_Name": "Skirball Cultural Center",
                "link": "link"
            },
            {
                "City_Name": "Los Angeles",
                "Museum_Name": "\nAutry National Center",
                "link": "link"
            }
        ]
    }
];

var new_data = [];
for (var state_index = 0; state_index < data.length; state_index++) {
    var state = data[state_index];
    var name = state.State;
    var cities = [];
    var city_index_map = {};
    
    for (var city_index = 0; city_index < state.Cities.length; city_index++) { 
        var city = state.Cities[city_index];   
        var museum = { "Museum_Name" : city.Museum_Name, "link" : city.link }
        
        var city_data = {};
        // City doesn't exist, so add it
        if (!(city.City_Name in city_index_map)) {
            city_index_map[city.City_Name] = city_index;
            city_data.City_Name = city.City_Name;
            city_data.Museums = [ museum ];
            cities.push(city_data);
        } else { // City already exists, so append the data
            var tmp_city_index = city_index_map[city.City_Name];
            cities[tmp_city_index].Museums.push(museum);
        }
    }
    new_data.push( {"State" : name, "Cities" : cities} );
}

var p2 = document.getElementById('json_clean');
p2.innerHTML = JSON.stringify(new_data);

want = [
    {
        "State": "California",
        "Cities": [
            {
                "City_Name": "Long Beach",
                "Museums": [
                    {
                        "Museum_Name": "Museum of Latin American Art",
                        "link": "link"
                    }
                ]
            },
    ...