First of all, Howard relies on Bernadette for shopping because he has no opinion. As can be seen from the TV series, he is a man without a definite view. Although he is also a brilliant scientist, every time he is ridiculed by Sheldon for his "low education" or research, he will be at a loss. I think he is at a loss, mainly because he doubts his ability. It can be seen that he has no opinion in his heart.
Just like when he was invited to space for the first time, he hesitated to go or not. If his friends hadn't supported him, Bernadette would have married him before he went to space, which reassured him. I think he would have been entangled in whether to go or not. Therefore, having no opinion is one of the reasons why he always consults Bernadette when shopping.
Secondly, Howard is a typical mambo type. He likes to listen to his mother's advice since he was a child. Even if you are dissatisfied with your mother's behavior, you will seek her help when you encounter problems. In the TV series, every time he goes home, he will report to his mother that I am back, even if his mother can't move, he will pick him up. And every time he leaves, he will let his mother know. On the one hand, this is a respect for his mother, on the other hand, he is the kind of little man who always needs maternal love. Need a sense of existence, which created his character of asking others for advice about everything he bought.
After marriage, it is easy to fall into the dilemma of mother-in-law and daughter-in-law. Bernadette is also the kind of woman who has both a "female man" and a "strong woman", but fortunately Bernadette is not the kind of unreasonable woman, although she is strong. So Bernadette and Howard's mother can still live in peace. Later, Howard's mother died. Of course, everything in Howard's life goes through Bernadette. Bernadette has to make decisions about small things, from house decoration to shopping.
Moreover, Howard's salary is not as high as Bernadette's, so under this gap, it is inevitable that Howard needs Bernadette to buy things. From the play, we can see that Howard relies heavily on his wife Bernadette in many ways. Even in scientific research, if there is any problem, he needs his wife to help him analyze. It can be seen that Bernadette's ability is above him.
Therefore, Howard must go through the consent of his wife's adult to buy things, so that in case he can't afford something one day, he needs to sell cute and rely on his wife to buy it for him.
Finally, I think that after a man like Howard gets married, the economic power must be in the hands of his wife Bernadette. Since there is no ticket in his hand, of course he has to ask his wife for instructions every time he buys something.
To sum up, it is not surprising that Howard relies on Bernadette for shopping.