@Margo - I have checked and the user is a member of the same security group that has access to the list - other people are not having this issue.
I have also given them direct edit access to the list - if that's been done will save only work for new items once the list has been refreshed or would it work for forms previously created but not yet saved?
Do you mean data in a not saved form?
I can't tell exactly whether the user will be able to submit an open form, as I'm not sure what access was missing. Have you checked if they can now submit the data?