I build CloudQuery which is able to turn most websites to API and it has a simple to use web interface to create the API. And it is open sourced on github

Current Solution As of Mid-April 2016 - Yahoo is Allowing YQL Requests Without Oauth Again Due to Developer Outrage

You can once again write a query without any authentication like the following:

https://query.yahooapis.com/v1/public/yql?q=select%20*%20from%20weather.forecast%20where%20woeid%20in%20(select%20woeid%20from%20geo.places(1)%20where%20text%3D%22nome%2C%20ak%22)&format=json&env=store%3A%2F%2Fdatatables.org%2Falltableswithkeys

Previous answers below in case they were of help for anyone. During certain periods of Yahoo's changes they did work.

Below are Older Versions of this Answer for Historical Reasons That May Also Still Work

Updated Answer After Yahoo's Latest Round of Updates - Insecure OAuth Workaround

You will need to create a Yahoo account & then create a web application at https://developer.yahoo.com/apps/create/

You will then need to use an OAuth Library to properly encode your Client ID & Client Secret. Here is an example in JavaScript based off a Yahoo Example Page & a 2008 Blog Article by Paul Donnelly. This generates an encoded URL to use to request a weather feed.

//Fill in your consumer Key & Secret from Yahoo's App & adjust location as needed. 
//This Key & Secret combination is invalid & won't work for you
var consumerKey = "dj0yJmk9NkRjbXpjUEhPbjlnJmQ9WVdrOVFUQTFaV2wxTjJrbXnHbz3NQSktJnM9Y29uc3VtZXJzZWNyZXQmeD0wOQ--";
var consumerSecret = "9bea8a9k3934d16365ek7e23e0abo1bba4q5c03c";
var locationToQuery = "90210"; //Can be zip code or anything that works in the query select woeid from geo.places(1) where text=<Your Location>


var makeSignedRequest = function(ck,cs,loc) {

    var encodedurl = "https://query.yahooapis.com/v1/yql?q=select%20*%20from%20weather.forecast%20where%20woeid%20in%20(select%20woeid%20from%20geo.places(1)%20where%20text%3D%22"+loc+"%22)&format=json&env=store%3A%2F%2Fdatatables.org%2Falltableswithkeys";

    var accessor = { consumerSecret: cs, tokenSecret: ""};          
    var message = { action: encodedurl, method: "GET", parameters: [["oauth_version","1.0"],["oauth_consumer_key",ck]]};

    OAuth.setTimestampAndNonce(message);
    OAuth.SignatureMethod.sign(message, accessor);

    var parameterMap = OAuth.getParameterMap(message);
    var baseStr = OAuth.decodeForm(OAuth.SignatureMethod.getBaseString(message));           
    var theSig = "";

    if (parameterMap.parameters) {
        for (var item in parameterMap.parameters) {
            for (var subitem in parameterMap.parameters[item]) {
                if (parameterMap.parameters[item][subitem] == "oauth_signature") {
                    theSig = parameterMap.parameters[item][1];                    
                    break;                      
                }
            }
        }
    }

    var paramList = baseStr[2][0].split("&");
    paramList.push("oauth_signature="+ encodeURIComponent(theSig));
    paramList.sort(function(a,b) {
        if (a[0] < b[0]) return -1;
        if (a[0] > b[0]) return 1;
        if (a[1] < b[1]) return  -1;
        if (a[1] > b[1]) return 1;
        return 0;
    });

    var locString = "";
    for (var x in paramList) {
        locString += paramList[x] + "&";                
    }

    var finalStr = baseStr[1][0] + "?" + locString.slice(0,locString.length - 1);

    return finalStr;
};

//Use the encodedURL to make your request
var encodedURL = makeSignedRequest(consumerKey, consumerSecret, locationToQuery); 

It should be noted never to show your consumer key or consumer secret to the public. You can use your own Yahoo Credentials in this Plunkr: http://plnkr.co/edit/EvLbgs

Original Answer

Unfortunately as of right now, the servers are down to create that app. Ideally, once they're back up you can use server side code to do the oauth part. I'll try to edit this answer when that happens. According to Yahoo the URL will be the same except without the /public part. The big difference will be that you need to send request headers with the URL that authenticate your account.

Here's a temporary fix until then. You can still use a YQL query geo.places with a zip code to get the woeid.

select * from geo.places where text=90210 limit 1

You can then grab your woeid from there & use it in the following url to get an xml feed:

http://weather.yahooapis.com/forecastrss?w=WOEID_GOES_HERE

I've created a Plunker as an example of this temporary fix here: http://plnkr.co/edit/dClPDtnToMhHqvKpfCzj?p=preview

Here's the gist of it though using jQuery:

var zipCode = 90210;

$.ajax({
    dataType: "json",
    headers:  { "Accept": "application/json; odata=verbose" },
    url: "https://query.yahooapis.com/v1/public/yql?q=select%20*%20from%20geo.places%20where%20text%3D"+zipCode+"%20limit%201&format=json&env=store%3A%2F%2Fdatatables.org%2Falltableswithkeys",
    beforeSend: function(xhr){xhr.setRequestHeader('Accept', 'application/json; odata=verbose');},
    success: function(data){
        $.getJSON("https://query.yahooapis.com/v1/public/yql?callback=?", {
            q: "select * from xml where url=\"https://weather.yahooapis.com/forecastrss?w="+data.query.results.place.locality1.woeid+"\"",
            format: "json"
        },function (data) {
          var weather = data.query.results.rss.channel;
          var html = '<div><span class="temperature">'+weather.item.condition.temp+'<span class="degree">&deg;</span><sup>'+weather.units.temperature+'</sup></span><br><span class="wind-chill">Feels like: '+weather.wind.chill+'<span class="degree">&deg;</span></span></div></a>';
          $("#weather").html(html);
        });
    },
});

From developer.yahoo.com:

"Important EOL Notice: As of Thursday, Jan. 3, 2019, the weather.yahooapis.com and query.yahooapis.com for Yahoo Weather API will be retired.  To continue using our free Yahoo Weather APIs, use https://weather-ydn-yql.media.yahoo.com/forecastrss. Contact yahoo-weather-ydn-api@oath.com for credentials to onboard to this free Yahoo Weather API service."

I have used the Yahoo Weather API XML format for years and noticed in that last couple of weeks this new bug. I tried to report the bug to https://developer.yahoo.com/weather/support but get a 404 page not found. I decided to parse the returned date if equal to current date to continue if not equal to re-call sub. this way I always get current weather but unfortunately that's a lot of unnecessary traffic / request maybe YDN will realize and fix. but without being able to report I don't know. I know this is not a fix but more a Band-Aid good luck!

Private Sub btnWeather_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles btnWeather.Click

    If InternetConnection() = False Then
        MsgBox("No internet connection!", vbExclamation, "Oops!")
        Exit Sub
    Else

        'MsgBox("Internet connection detected!", vbInformation, "Huray!")

        btnWeather.Enabled = False
        lblWorking.Text = "Working ..."
        tbTries.Text = "1"

        Try

            Dim t As New clsWeather(Format(Me.TxtBoxZIP.Text), "f")
            lblTodaysDate.Text = FormatDateTime(Now.Date, DateFormat.ShortDate)
            tbHigh.Text = t.high & "°"
            lblCity.Text = TxtBoxZIP.Text & " Weather "
            tbLow.Text = t.Low & "°"
            tbDay.Text = t.day
            tbDate.Text = t.date1
            tbCurrenttemp.Text = t.currenttemp & "°"
            tbCurrentCode.Text = t.currentcode
            tbForcastCode.Text = t.ForcastCode
            tbSunrise.Text = t.Sunrise
            tbSunset.Text = t.Sunset
            tbWind.Text = CInt(Val(t.Wind)) & " mph"
            tbHumidity.Text = CInt(Val(t.humidity))
            imgWeather.Image = Image.FromFile(t.GetImage)
            CodeName()


            If t.currenttemp < 85 And t.currenttemp > 45 Then

                lblFeelsLike.Text = ""
                tbFeelsLike.Text = ""

            End If

            If t.currenttemp > 85 Then

                lblFeelsLike.Text = "Heat Index:"

                Dim Temp = t.currenttemp
                Dim RH = CInt(Val(t.humidity))

                tbFeelsLike.Text = (-42.379 + 2.04901523 * Temp) + (10.14333127 * RH) - (0.22475541 * Temp * RH) - (0.00683783 * Temp * Temp) - (0.05481717 * RH * RH) + (0.00122874 * Temp * Temp * RH) + (0.00085282 * Temp * RH * RH) - (0.00000199 * Temp * Temp * RH * RH)

                Dim num As Decimal = CType(tbFeelsLike.Text, Decimal)
                Me.tbFeelsLike.Text = String.Format("{0:n0}", num)
                tbFeelsLike.Text = tbFeelsLike.Text & "°"

            End If

            If t.currenttemp < 45 Then

                lblFeelsLike.Text = "Wind Chill:"
                tbFeelsLike.Text = CInt(Val(t.Chill)) & "°"

            End If


        Catch ex As Exception
            MsgBox(ex.Message)
        End Try

    End If


        Dim day As String = DateTime.Now.ToString("dd")
        If day = tbDate.Text = True Then
            tbDate1.Text = tbDate.Text
            btnWeather.Enabled = True
            lblWorking.Text = ""
        Else
            btnWeather_Click(sender, e)
            tbTries.Text = tbTries.Text + 1
        End If


End Sub