I wonder if there’s a misconfiguration in your server’s PHP, or if the server lacks the mechanism to send mail.
eform uses the phpmailer class to send mail. While it is a fairly capable mail library (it can send mail via 3 different mechanisms - SendMail, SMTP or PHP’s native mail function), eform only utilizes the PHP native mail mechanism. I have a few server configurations where PHP can’t send mail using it’s default mail function. This can occur for various reasons. I’m not a mail expert at all, so I can’t go into detail about it. But I do know that on these servers, when I send mail from my php code, I have to use a library like phpmailer or Swift to send via SMTP. Then it works fine.
You could try running a little php script like this to isolate and test php’s native mail function:
If you put that script on your server, call it "mailtest.php" or something, then call it up in your browser, you should see the script output the number 1 and nothing else (and you should get the test email delivered to the email address in the script). If it outputs 0 or anything other than 1, then the mail function did not work due to configuration or something.