Hi,
Ok, so I just changed over to the iCloud calendar and sent my husband some calendar invites from various places (from Firefox, from my iPhone calendar app, from a Windows mail app and from Outlook). I really like the online iCloud calendar and that is what I use the most. He uses gmail. He was not receiving the invites for the most part except the ones from the mail app.
So after much testing, it turns out it was because of an Apple ID which he really doesn't use, and he didn't remember setting up an iCloud account, so this was not obvious to us! He has a older, rarely-used iPad, hence the iCloud account. It is tied to his gmail account (apple ID). Turns out his iCloud calendar online setting needed to be changed. Even though he had never logged into iCloud before, I asked him to please check anyway to see if he had an iCloud account and see if the calendar was by chance defaulted to "receive in-app notifications." Yes, it was, and there was a stack of the missing invitations from me right there on the calendar page notifications inbox. We changed this to the other option (Settings, Advanced, Invitations, Receive event notifications as email). Happy to say he now he receives calendar invites from me when I send them!
I could see how a lot of other folks might experience this if people have an iCloud account they have abandoned or are somewhat 'unaware' of because they might have had an iPhone or iPad in the past, for example, and this same thing might have occurred.
Anyway, I hope this helps someone. : )