If Mathmom is right – and she probably is – then a kind mom would help the kid to write – or would actually write – a sample thank you note for the kid to copy, saying that she (the kid) was sorry that she was too sick to attend the party but is extremely grateful for all the love and support and for the gift.
If extreme anxiety or phobia were the issue, the mom could actually type up a generic note, print out a dozen copies, and sit with the kid as she signed them.
It goes without saying that therapy seems in order.