Firefox enables spell check in text area fields by default, but not input fields. Here is a quick way to fix that.
 
1. Type about:config into the address bar
 
2. A warning message will appear. Click the I’ll be careful, I promise! button.
 

 
3. Enter layout.spellcheckDefault in the bar at the top of the about:config page.
 

 
4. Double click the preference layout.spellcheckDefault and change the value to 2.
 

 
5. Click Ok to confirm the change.