When my children were growing up, they were given an allowance (that is, pocket money) from about your daughter's age. The amount was so small it would probably shock you. The reason they were given an allowance was because we wanted them to learn how to use money properly (and we structured the use of that money), but we didn't want them to believe that they were entitled to payment for any little thing they did at home.
Making their beds, picking up their own toys, setting the table, feeding the pets, etc., were jobs they did because they were part of the family, not because they were being employed by us to do them. I did make charts so they could see what they needed to do and so they could check the jobs off (which is itself a basic form of reward) as they did them. When they grew older, they were able to *work* for money, both inside and outside the home. There were extra jobs they could do, for pay, when they were old enough and capable enough.
I see the other side of it, too. My older son and his wife have their children earn their allowance - in the sense that, if they don't do what they're expected to do at home, the money is not forthcoming. He says that since it works that way in the business world, it can work in their home. It seems to be successful; the parents are on the same page about it, and the girls are pretty good about their responsibilities.
Is, perhaps, your husband thinking of allowance/reward as a type of bribery to get a child to do what she should do anyhow? If so, that's worth discussing. It *can* turn out that way - when the child starts taking the upper hand and saying, "You want me to do what? What'll you give me if I do it?" But that can be easily dealt with. That's an attitude thing.
Rewards can be useful. Eventually your daughter will learn how to reward herself. (It takes quite a while, though, for "I did my work well" to be a reward.) Rewards, too, are capable of getting out of hand. That's why it's necessary for the Mom-and-Dad Team to sit down and talk it through. "What's our long-range goal in all this? What do we want to teach? How can we teach it best right now?"