Sure, it's quite easy. You add a form field button with a Submit Form action as its Mouse Up event, and as the URL you enter this (using your own email address, of course):
mailto:me@server.com
Clicking this button will first trigger a validation process of all the fields which are set as required, and the file won't be submitted unless they're all filled-in.
The probable reason your file doesn't work in Firefox is that it's using its own PDF viewer plugin (Chrome and Safari have ones as well), instead of the Adobe plugin, and that plugin is not very good when it comes to form fields. Regarding IE it's more difficult to say without additional information.
You should follow these instructions to make sure you're using the Adobe plugin in all of your browsers:
http://helpx.adobe.com/acrobat/using/display-pdf-browser-acrobat-xi.html