Pure Javascript with WebRTC

This is based on https://github.com/diafygi/webrtc-ips. You can copy & paste the example below into your Dev console to see it in action.

Firefox and Chrome have implemented WebRTC that allow requests to STUN servers be made that will return the local and public IP addresses for the user. These request results are available to javascript, so you can now obtain a users local and public IP addresses in javascript. This demo is an example implementation of that.
Additionally, these STUN requests are made outside of the normal XMLHttpRequest procedure, so they are not visible in the developer console or able to be blocked by plugins such as AdBlockPlus or Ghostery. This makes these types of requests available for online tracking if an advertiser sets up a STUN server with a wildcard domain.

//get the IP addresses associated with an account
function getIPs(callback){
    var ip_dups = {};
    //compatibility for firefox and chrome
    var RTCPeerConnection = window.RTCPeerConnection
        || window.mozRTCPeerConnection
        || window.webkitRTCPeerConnection;
    var useWebKit = !!window.webkitRTCPeerConnection;
    //bypass naive webrtc blocking using an iframe
    if(!RTCPeerConnection){
        //NOTE: you need to have an iframe in the page right above the script tag
        //
        //<iframe id="iframe" sandbox="allow-same-origin" style="display: none"></iframe>
        //<script>...getIPs called in here...
        //
        var win = iframe.contentWindow;
        RTCPeerConnection = win.RTCPeerConnection
            || win.mozRTCPeerConnection
            || win.webkitRTCPeerConnection;
        useWebKit = !!win.webkitRTCPeerConnection;
    }
    //minimal requirements for data connection
    var mediaConstraints = {
        optional: [{RtpDataChannels: true}]
    };
    var servers = {iceServers: [{urls: "stun:stun.services.mozilla.com"}]};
    //construct a new RTCPeerConnection
    var pc = new RTCPeerConnection(servers, mediaConstraints);
    function handleCandidate(candidate){
        //match just the IP address
        var ip_regex = /([0-9]{1,3}(\.[0-9]{1,3}){3}|[a-f0-9]{1,4}(:[a-f0-9]{1,4}){7})/
        var ip_addr = ip_regex.exec(candidate)[1];
        //remove duplicates
        if(ip_dups[ip_addr] === undefined)
            callback(ip_addr);
        ip_dups[ip_addr] = true;
    }
    //listen for candidate events
    pc.onicecandidate = function(ice){
        //skip non-candidate events
        if(ice.candidate)
            handleCandidate(ice.candidate.candidate);
    };
    //create a bogus data channel
    pc.createDataChannel("");
    //create an offer sdp
    pc.createOffer(function(result){
        //trigger the stun server request
        pc.setLocalDescription(result, function(){}, function(){});
    }, function(){});
    //wait for a while to let everything done
    setTimeout(function(){
        //read candidate info from local description
        var lines = pc.localDescription.sdp.split('\n');
        lines.forEach(function(line){
            if(line.indexOf('a=candidate:') === 0)
                handleCandidate(line);
        });
    }, 1000);
}
//log IP addresses
getIPs(function(ip){
    //local IPs
    if (ip.match(/^(192\.168\.|169\.254\.|10\.|172\.(1[6-9]|2\d|3[01]))/))
        console.log("Local IPs: "+ip);
    //IPv6 addresses
    else if (ip.match(/^[a-f0-9]{1,4}(:[a-f0-9]{1,4}){7}$/))
        console.log("IPv6 Addresses: "+ip);
    //assume the rest are public IPs
    else
        console.log("Public IPs: "+ip);
});

With the ipdata API

The advantage of using our API is you can expect this method to work reliably across all browsers and it is super-fast. We run out of 11 AWS data centers around the world for millisecond-optimized performance.

$.get("https://api.ipdata.co?api-key=<<apiKey>>", function(response) {
    console.log(response.ip);
}, "jsonp");