Is zakat obligatory on capital borrowed from a relative who does not demand its repayment, given the presence of a partner to manage the project and the allocation of profits for household expenses, or must the debt be repaid first?

Zakat (definition) is obligatory on vegetables prepared for sale because they are trade goods. Thus, Zakat is obligatory on your share of them if their value reaches the nisab (definition) when combined with money or other trade goods. As for debt, it is not deducted from the wealth subject to Zakat if you possess trade goods that you can dispense with to cover the debt. Otherwise, the debt is deducted, and you pay Zakat on the remainder if it reaches the nisab.