This sort of tulip pisses me off.
I usually go into a quiet room with the most senior person that willing to listen to me and tell them that it is inappropriate to fill in an employee holiday card since:
I am clearly not an employee of your company
I have no contract of employment with you company
I do not get paid holiday from your company
I do not receive a holiday allowance from you company
I have a contract of employment and annual leave allowance with <your company name here>
Therefore to fill it in would be absurd.
I also mention (some IR /agent style FUD) that if I were to fill in the holiday card it might make life difficult for the client if someone were to make employment rights claim or sue for unfair dismissal or request holiday pay etc.
As other have said offer to give copies of <your company name here> holiday cards that they can hold for their records.
Keep resisting this tulip. FFS - WHEN WILL CLIENTS LEARN!!!!
PS - I also mention I have tax investigation insurance that forbids me from signing any form as an employee or temp worker for any client, otherwise I lose cover. (I make this bit up).