YANBU - how awfully for your DD.
I come from a family where my Mum NEVER, and I mean NEVER forgets a birthday, anniversary etc. She is like a sergeant major - prepared for any celebration . However, my sister once rang me, on my birthday, to moan about her life, asked me what I was up to and when I told her, well I'm celebrating my birthday, was gobsmacked that she had forgotten . We laugh about it as she is usually very organised too (coming from the same mother, we have NO choice but to be!!!)
On the other hand, my MIL puts absolutely no importance on birthdays and feels Christmas should be reserved for Jesus - to be fair she is old school Italian Catholic so this is major in her life. HOWEVER, she has never forgotten my DS's birthday and feels that, whatever her feelings on birthdays, Xmas or any other celebrations, my DS, her DGS, deserves to have a fab time on them all.
So, I would be incandescent with rage, like you. I would want to send a majorly stinging shitemail but would proably be talked down by my DH. I would, however, call them to ask when they put the card etc in t he post as you are terribly worried that it has been lost by the Post Office and you would hate for their hard-saved pensions to be wasted like this If they have the cohones to say they didn't send anything, I would blatently ask 'why'? Ask what their reasoning is that a child would be able to understand why their GP's/Aunt etc would have not thought them important enough to send something to.
Sorry for the length of this reply - but this is a bit of a bugbear for me...I have a really hard time accepting that there are people out there who, for their own weird reasons, apply their own {odd} values to children.
I really don't care if I don't get a card/pressie etc, I am an adult, but not children, they should always be remembered.
Do something wonderful with your DD.