Hi McUsher,
The article Add a member to a Family Sharing group on iPhone specifies that invitations can be sent via AirDrop, Text Message or Email. Are you able to test AirDropping that invitation? Can you successfully send other messages and emails to that user?
If other messages are not working either, If you can't send or receive messages on your iPhone or iPad outlines a series of troubleshooting steps to use which may help correct that problem:
"If you see a red exclamation point
If you try to send a message and you see
with an alert that says Not Delivered, follow these steps:
- Check your network connection.
- Tap
, then tap Try Again. - If you still can’t send the message, tap
, then tap Send as Text Message. Messaging rates may apply. Contact your carrier for more information.
You can set up Messages so that it automatically tries to send messages as an SMS when iMessage isn't available. Go to Settings > Messages, and turn on Send as SMS."
Regards.