All that button does is create what's called a mailto link with the website link in it. That mailto link is handled by whatever program is set to handle that in the Applications section of the Firefox settings.

If it's an external program, that program should have the option for you to select the email address. If it's a webmail program (like Gmail), it's going to open up in whatever email address you are logged into.

Thanks Wesley, you were right, I fixed it!