Of course I run the test at the wppizza plugin, this is the output:

Array
(
[error] => FAIL - check details below
[error-info] => Array
(
[last-error] => Array
(
[type] => 2
[message] => mail(): SMTP server response: 503 This mail server requires authentication when attempting to send to a non-local e-mail address. Please check your mail client settings or contact your administrator to verify that the domain or address is defined for this server.
[file] => C:\Inetpub\vhosts\XXXXXcensoredXXXXX\wp-includes\class-phpmailer.php
[line] => 698
)
[phpmailer] => Could not instantiate mail function.
)
[debug] => 
)

I am not using another plugin related with SMTP.

As normal test order this is the error:

EMAIL TO SHOP FAILED: Could not instantiate mail function. | Array
(
[type] => 2
[message] => mail(): SMTP server response: 503 This mail server requires authentication when attempting to send to a non-local e-mail address. Please check your mail client settings or contact your administrator to verify that the domain or address is defined for this server.
[file] => C:\Inetpub\vhosts\XXXcensoredXXX\wp-includes\class-phpmailer.php
[line] => 698
)
| Recipients: [email protected] | Subject: [Pizza XXX] - ORDER TEXT 27. February 2018 15:19 | Headers: Array
(
[0] => Content-Type: text/html; charset=UTF-8
[1] => From: Pizza XXX <[email protected]>
[2] => Reply-To: [email protected]
[3] => Cc: [email protected]
)

What do you mean with “verify that the domain or address is defined for this server.”?
The wordpress installation runs at subdomain.MY-DOMAIN.com and the email is [email protected]

But the normal contact form like Contact Form 7 is working without a problem? Could that be nessessary?

Ok I solved it. Just change in the file /wp-includes/class-phpmailer.php the paragraph public $SMTPAutoTLS = true; to public $SMTPAutoTLS = false;

Okay, so do you know how to solve this without changing core files?