First suggestion, have them open up a ticket with cPanel if they can't fix it 
Next suggestion, it's likely they either didn't restore or restored with the wrong permissions the /home/user/etc/domain.com folder (that folder contains the individual email account logins in passwd and shadow files). If any of those files have the wrong permissions or are missing the email account users, then you won't be able to log into webmail:
/home/user/etc permissions
Code:
drwxr-x--- 6 user mail 4.0K Jan 28 15:27 etc
/home/user/etc/domain.com permissions
Code:
drwxr-x--- 5 user mail 4.0K Jul 9 12:50 domain.com
/home/user/etc/domain.com/ permissions inside the folder
Code:
# ls -lah
total 44K
drwxr-x--- 5 user mail 4.0K Jul 9 12:50 .
drwxr-x--- 6 user mail 4.0K Jan 28 15:27 ..
drwx--x--x 4 user user 4.0K Apr 10 15:52 admin
drwx--x--x 3 user user 4.0K Apr 10 15:52 cpacct
-rw-r--r-- 1 user mail 152 Feb 24 13:23 passwd
-rw-r--r-- 1 user user 1.9K Feb 24 13:23 passwd,v
drwxr-x--- 2 user mail 4.0K Feb 24 13:24 @pwcache
-rw-r--r-- 1 user user 33 Feb 24 13:23 quota
-rw-r--r-- 1 user user 387 Feb 24 13:23 quota,v
-rw-r----- 1 user mail 107 Apr 10 16:00 shadow
-rw-r----- 1 user user 2.0K Feb 24 13:23 shadow,v
Contents example of passwd file at /home/user/etc/domain.com/ location:
Code:
# cat passwd
admin:x:510:510::/home/user/mail/domain.com/admin:/usr/local/cpanel/bin/noshell
cpacct:x:510:510::/home/user/mail/domain.com/cpacct:/user/local/cpanel/bin/noshell