He's got an antiquated attitude, unfortunately.
You should not be "asking" for money. It's not his. It's family money. You should each have a certain amount of discretionary money that you can each spend without discussing it with the other, and the rest goes to household expenses and things you jointly decide on.
Can you discuss other things with him in a reasonable way - is it just money that is his stumbling block? Or is he domineering in other ways? Does he belittle you? Does he try to make you feel stupid? It sounds like nothing you want to do is valued - if vacation is only with his family.
The solution that some have suggested is that you get a job. Maybe. But if you don't make the same amount as he does, won't you still have the same problem? And if you make more, he won't handle it well, right?
If money is coming out of your account but he's not putting money in to cover it, and if he won't switch them to his account, it sounds like a control issue big time. Are you on both accounts? If so, switch the bills over to HIS without asking him, and then get him off your account. If both accounts are at the same bank, you can sign up for on line banking, transfer money from his account into yours, but he can't do the reverse. I know this will make him angry, but it's financially irresponsible for him to allow you to incur overdraft charges or returned check fees on FAMILY MONEY.
The problem sounds like he doesn't value what you do. There are supposed to be some great websites to calculate the costs of day care, chauffeuring, tutoring, cooking, being a personal shopper, catering for family gatherings, trip planning, taking people to medical appointments. Be sure to add up what you do, and add time & a half for weekends and double time for holidays. Factor in vacation time and sick time which you don't get. Present the whole thing to him and let him see what it would cost to replace you!
The other thing to do is to take a weekend away and leave him with his own children. Go see YOUR family or go for a ladies' weekend, anything. If you don't have the money, can you borrow it from your family? Get out of there for 48 hours and let him cope! He may have new respect for what you do.
If you don't think he's teachable, then you may have to consider whether this is a fulfilling relationship you want to stay in.
Go for counseling. If he won't go, go without him. Work it through with an objective person. Your physician can refer you to someone who accepts whatever medical insurance you have.
This situation is not sustainable the way it is. But no one can take advantage of you without your cooperation. So find a way out.