Mom bought the cards. She kept a box of them, a book of stamps in the junk drawer, and a list in the front of her address book that told her whose birthday was coming. Somewhere along the way, the cards became one more thing that happened without anybody deciding it would.
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You may notice it first from the outside. A grandchild's birthday passes with no card in the mail. A friend sends a note and nothing comes back. Your father still knows when everyone's birthday is, he just never gets around to writing it down, and by the time he does, the day is gone.
How to tell the cards have stopped
Check the drawer where the stamps lived. If it's empty, or the stamps are the old ones, he has stopped buying them. Look at the fridge: is the birthday list still there, with the old birthdays still on it and nobody updating it? Ask the people who used to hear from him. Some will tell you outright that nothing has come since the funeral, and they may be too polite to say it to him.
What to do tonight
Don't make a project of it. Ask him one question: who should we send a card to next? Sit with him at the table with the address book, the stamps, and a pen. Let him write the message while you address the envelope, or the other way round. He'll likely complain that your handwriting is bad. Let him.
Then buy one box of cards with a few blank ones, and put it where she used to keep hers. Move the birthday list under a magnet on the fridge so it stays in sight.
What to set up so it keeps going
Pick one day a month for cards, and make it his day. Write the birthdays for the next three months in a single column so you can see what's coming. A missed card is not a disaster, since a late card still lands. Nobody is keeping score except the grandchildren, and mostly they just want to see their name on an envelope.
When to ask the doctor
If the cards are one thing on a list of things that are slipping, such as missed bills, repeated questions, getting turned around on a route he knows, or losing the thread of what he was doing, write those down with what you saw and when you saw it, and bring the list to his doctor. Don't wait for a crisis. The doctor needs to hear it from you in plain words, and you are allowed to ask for a check-up for that reason alone.