Greetings Lucy84,
Thanks for using the Apple Support Communities! We understand that you are having some troubles inviting members to your Family Sharing. Let’s take a look and see what we can do. We're happy to help!
We were able to gather a great resource to help invite family members to your Family Sharing account. Try to invite using their phone number and/or their Apple ID email. If one doesn't work, please try the other.
Set up Family Sharing
If you continue to experience any troubles inviting via iMessage, you can try to "Invite in Person" with the individual typing in their Apple ID and password on your device. To do this you will need to invite them from an iPhone, iPad, or iPod touch that is signed into your Apple ID. To do this you will open "Settings" > tap [Your Name] > tap "Family Sharing" > tap "+ Add Member" > select "Invite in Person." This will ask for their Apple ID and password.
Have a great day!