Emails not being received by some people
Solution 1:
Ideally, you need to check the logs if possible to see what is going on. This will give you a complete break down of each 'stage' of the email process and will hopefully provide extra information.
It also sounds like you're using a third party to send the email, if this is the case, then there is nothing we can do as it could be a bug in their system, for example.
Since your comments mentions it could be just delayed, read up on grey listing
Greylisting (or graylisting) is a method of defending e-mail users against spam. A mail transfer agent (MTA) using greylisting will "temporarily reject" any email from a sender it does not recognize. If the mail is legitimate the originating server will, after a delay, try again and, if sufficient time has elapsed, the email will be accepted.
And most intersetingly
Greylisting is effective because many mass email tools used by spammers do not queue and reattempt mail delivery as is normal for a regular Mail Transport Agent
Solution 2:
You can do some manual troubleshooting as follows:
You can find out what the server should be for any domain by looking up the "MX records"
For example here: http://mxtoolbox.com/
The MX record is what your email server looks for to find out where they should send your email.
You can then initiate a manual connection to the server listed in the MX record to see if it is accepting email and what error messages you might get. Use a telnet program like Putty: http://www.putty.org/ and telnet to the email server on port 25. Some of the commands you will need are listed here: http://www.yuki-onna.co.uk/email/smtp.html
So now you can connect to their mail server and send an email using your email address as the "From" address and see how the server responds directly. Try it from your computer and also from the server and see how their email server responds to the different IP addresses. Any email error codes that are returned can be looked up in google or here: http://www.serversmtp.com/en/smtp-error
Once you have checked that you can connect to the server it may tell you why your email is being rejected as spam or for some other reason, but the reason may not be easy to decipher. At this stage I would suggest you ask your customer to contact their support number with the error codes (or lack of them) that you received from their server. You can't log a problem yourself because the problem is on the receiving server. The customer will have to log that problem themselves. If they are not able to get the settings changed on their email server you may be able to find out what you would need to change on your server to get your emails delivered.
Some possible problems are your server being listed on a block list or lacking a SPF record and you can fix that yourself and check using various online validation services. For example https://www.mail-tester.com/