How can I programmatically take a screenshot of a webview, capturing the full page?

Try this one

import java.io.FileOutputStream;

import android.app.Activity;
import android.graphics.Bitmap;
import android.graphics.Canvas;
import android.graphics.Picture;
import android.os.Bundle;
import android.view.Menu;
import android.webkit.WebView;
import android.webkit.WebViewClient;

public class MainActivity extends Activity {

    WebView w;

    @Override
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        w = new WebView(this);
        w.setWebViewClient(new WebViewClient() {
            public void onPageFinished(WebView view, String url) {
                Picture picture = view.capturePicture();
                Bitmap b = Bitmap.createBitmap(picture.getWidth(),
                        picture.getHeight(), Bitmap.Config.ARGB_8888);
                Canvas c = new Canvas(b);

                picture.draw(c);
                FileOutputStream fos = null;
                try {

                    fos = new FileOutputStream("mnt/sdcard/yahoo.jpg");
                    if (fos != null) {
                        b.compress(Bitmap.CompressFormat.JPEG, 100, fos);

                        fos.close();
                    }
                } catch (Exception e) {

                }
            }
        });

        setContentView(w);
        w.loadUrl("http://search.yahoo.com/search?p=android");
    }

    @Override
    public boolean onCreateOptionsMenu(Menu menu) {
        getMenuInflater().inflate(R.menu.activity_main, menu);
        return true;
    }
}

Add INTERNET PERMISSION and WRITE_EXTERNAL_STORAGE in AndroidManifest.xml file.

Need to ask permission at run time for file write if app is running on or above Marshmallow.

webview.capturePicture() is deprecated, use below way can do:

Bitmap bitmap = Bitmap.createBitmap(webview.getWidth(), webview.getHeight(), Config.ARGB_8888);
Canvas canvas = new Canvas(bitmap);
webview.draw(canvas);
return bitmap;

@avinash thakur codes actually works, you just forget to mention to add <uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE"/> at Android Manifest file, otherwise, android wont allow your stream to write data / file on device.

cheers.