The most common reason why the contact form does not work is that it will not send an email to the same email the email is being sent from. Meaning, if you have email@example.com set up in Settings/General, and are inputting firstname.lastname@example.org when filling out the contact form, it will not send the email you're trying to send. You will need to try a different email to test the functionality of the contact form.
If that isn't the issue, then I would recommend checking your spam/junk folders to see if they are ending up there. If not, and you're still having trouble, let me know as I'd be happy to verify the store to take a look.