I am curious when CAR's are necessary - is it at every fault/error? <snip>
No. Just
NO.
Are you specifically interested in the scenario you presented, or are you asking "in general"?
If "in general", there are quite a few existing discussions here both recent and over many years.
Including some interesting conversations, such as:
How to deal with too many CARs (Corrective Action Requests), PARs (Preventive Action)
And check this:
Corrective Action Requests - Tagged Discussions for some thoughts.
Always think about the specific scenario before you issue a Request for Corrective Action. If you don't, it will reduce the importance of corrective actions in your company (in part by alienating employees) and you could end up with hundreds of "non-essential" corrective actions which are not actually fully addressed.
In the scenario you mention, the first thing to do is find out why it happened, and if it happens often (recurrence). You can do this through during the audit by asking questions and looking at records. Discuss with the auditee.
Get the facts during the audit.
As to emails, everyone should know that emails are not reliable at times. I guess you could put this in the "software validation" bucket. Have emails been sent and confirmed as received a few times?
NOTE: From time to times email filters will put an email from a "valid" sender which is typically accepted into a spam folder. The only way to prevent this is to "white list" that sender, and even that isn't always a "guaranteed" solution. The best thing is to be on an email system which supports "Return Receipt" to indicate an email has not only been received, but also whether it has been read.
Personally I wouldn't consider your scenario as significant as long as your company is "understanding" with respect to a supplier not paying on time in such a case (their not receiving an emailed invoice). Yes, it is how much business is done today (invoice by email) and I'm not saying it's not appropriate. I'm only saying that the failure modes and effects of email must be understood.