Would the wife bear a burden of sin for her husband not paying Zakat on the gold he bought for her, knowing that she does not work and has no income, and fears paying Zakat from the household expenses due to disputes between her husband and his family?

There is no zakat on the house and the car. As for cultivated land, the zakat on the harvest (half of one-tenth for irrigated crops, and one-tenth for non-irrigated crops) is obligatory if it reaches the nisab, and it is obligatory on staple grains and fruits that are measurable, such as barley, wheat, and dates. There is no zakat on vegetables and fruits unless they are sold, in which case zakat becomes obligatory on their price if a hawl (lunar year) passes over it and it reaches the nisab. If the husband rents out the house, car, or land, zakat becomes obligatory on the rent if a hawl passes over it and it reaches the nisab. If he owns other money and it reaches the nisab and a hawl passes over it, its zakat is obligatory on him, and if he does not pay it, he is sinful, and there is no sin on the wife. It is not permissible for the wife to pay zakat on behalf of her husband except with his permission and intention. The nisab for banknotes and what is equivalent to them is the equivalent of 85 grams of gold or 595 grams of silver, and the amount of zakat is a quarter of one-tenth (2.5%). There is no zakat on ornamental jewelry according to the majority of scholars.