I'm sorry to hear money is tight & I hope things improve for you.
Here is my opinion
I wouldn't put money in the piggy unless you could be certain it will stay put. (for me) The idea of a piggy bank would be to teach a child the concepts of money management. Putting it in for savings & then pulling it right back out doesn't really teach them responsibility.
I have an account already for both my boys. We keep a log book for them. They are free to open & free to maintain so why not? It's cute to see a written log of gifts, such as $10 from grandma for my first birthday.
I'm a much older mom, so money is no longer tight for us. However, I have been through some very tough times.
Just keep what you need out of the money they get for gifts & put a tiny bit of it aside for them. Banks don't care if you take them in to deposit $2. Kids get a kick out of going to the bank. Then let it stay put.