This model can send and print emails. It cannot download them, it only reads them from the server and then prints (so it's not viewable on the LCD of the printer for example)!
If your email is not on the server, then there is nothing to print.
Also, it prints ALL emails on the server, you cannot specify which one(s).
My guess is you're using a POP3 client (like Outlook/Thunderbird) and you're downloading all emails to your machine and then removing them from the server (this is automatic and can be configured in your email client) meaning no emails are left on the server.
To print all emails on the server (just for your email account obviously):
Ensure your firmware/drivers are up to date.
Look under Protocols section under the Network tab.
More details (most of which you know, but I include all for fullness):
After updated, log into the Brother machine's web management
use 'access' as the password
click on the network tab and then protocols
select pop3 / smtp
Set your pop 3 info, address, mailbox, pass etc...
Select the FAX tab
Select email receive
Update the poll time.
If there are any emails on the server, it will automatically print them all at that time. It will not delete or touch your emails, so, if your email was not removed from the server and your poll was set to 10 minutes, it would mean the same email would print every 10 minutes.
If this is not the case, you will need to talk to Brother directly.
More details about why POP is used on your machine
POP3 can be used for "Pop before SMTP" (which is just an authentication method and has nothing to do with downloading emails)
Read your manual page 55 for more detail or read this generic Brother manual