www.wysiwygwebbuilder.com

I am using a contact form on my website which seems to work fine and when I have sent tests they have gone through fine to everyone. We have had a person however which claims they have sent a message or two which we didn't receive. 

That started me thinking. When we send an e-mail via an e-mail programme, if it doesn't go through, we get an e-mail from the server saying that the e-mail could not be sent. What happens to these warning e-mails when we are using a website contact form?

If you use the built-in form script then the email is sent by the server (= your web host).
If something goes wrong then this usually means that there is some kind of filter on the server side (for example because it does not allow certain email addresses) or the email ended up in spam.

So if the messages doesn't go through for some reason, the sender will be completely unaware?

Correct, because there is no way to know if the server has sent the email.
However, if it fails then it will always fail for certain email addresses, so this is something you can test and communicate with your web host.
Of course, you cannot prevent emails going to spam.

When using Outlook, if a message doesn't go through I get a message titled like this "Mail delivery failed: returning message to sender". Do we not get a similar e-mail when using a form? My contact forms go to people using with several different domain e-mails.

No, unless it is a feature provided by your host, it has nothing to do with the form script.