You can check support for speech recognition in various browsers at Can I Use. At the moment only Chrome supports speech recognition reliably. Others don't, including IE, Firefox and Safari. Firefox is working on the speech recognition support, but it's not there yet.
If your browser supports getUserMedia to record audio you can record the audio and send it to the server for recognition. You can also use javascript-only recognizer Pocketsphinx.js if you want to recognize few simple commands.
On browsers which do not support getUserMedia API you can use Flash object or java applet to capture audio. That's pretty much your only option if you prioritize IE.