Just playing around with Firebug and I noticed that my form fields have a "spellcheck" field. Of course Firefox doesn't spell check plain text fields normally, so it's set to false except in text areas. The cool part is that this is just a plain attribute. Feel free to turn it on, like this:
document.getElementById("myTextInputField").spellcheck = true;
Then start typing in that text input field and Firefox will spell check it as you type.