Should I send a blessing message to my daughter-in-law first on holidays?

There is no fixed pattern for this. If you think you should send it, you don't have to if you feel busy. It doesn't affect any essential problems, but it's best to send it out of courtesy, because it will show that you care more about her. There is no need for such frequent contact between daughter-in-law and father-in-law, but proper greetings are still needed.