Your husband is the one creating the rift between you. Regardless if how much pressure his mother or sister pit him under, he doesn't have to go along with it. He chooses to. He chooses to deprive you and your children (the family he created, who are his primary family now, not his birth family) because he prefers to disrespect you in this way rather than tell them no.
Take out a bank account in your sole name. Have your wages paid in there. Pay no more than half (depending on what you earn versus what he earns - it needs to be a fair proportion and he likely earns more, men usually do because they don't take maternity leave affecting their career) towards family expenses, which includes child related expenses. The rest left over is your money to spend as you wish.
He can do the same. If he is broke from giving all his spare money to his sister and mother, don't you pay for him to go out or on holiday or whatever, and don't you pay his share of the family finances. Otherwise he is literally stealing from you and you would be going to work for no reason, if his mother and sister gets your wages. Giving money to them must be considered his personal hobby, his choice, and the money must not come from that which is needed for family expenses or from your own wages at all.
This is what I would do if I was staying together with him. I would also divorce him though, because that's the only way to truly separate finances. Otherwise if you divorced years from now, he may have spent all his money on his mother and sister, while you may have amassed savings. Because those savings amassed whilst married, they'd be a marital asset and he could claim half in the divorce. So I would separate finances with a divorce before amassing any savings.
Honestly though, I wouldn't really want to stay with someone who thought it was ok to steal from me and give our family's money to his mother and sister. That is someone with very low morals and as someone with high morals myself I'd consider it to be a mis-match. I need to be in a relationship with someone who is my equal, not someone who behaves in an inferior way and disrespects me.
There is no point in writing to your MIL or SIL because they are horrible people. Asking horrible people to not be horrible, never works. They have no shame and don't care about what you think or feel. You can't appeal to their "better nature" when they don't have one! If you get divorced or break up, don't keep it a secret from people the reason why. You don't owe your husband that protection from his own shame. Let others think badly of all of them, it's not your problem. If they don't like it then they should have thought about that before carrying out their actions. You don't owe them protection from themselves. Don't feel bad for them. They don't even think you're owed your own wages which you earn! They're despicable.