Is it permissible to demand profits for a sum paid to purchase an apartment that has not been received due to a dispute between partners, given the offer to either refund the amount or wait, and knowing that the seller is a merchant whose money is invested in trade?

If you paid the price of the apartment to the seller to complete its construction according to the agreed specifications and time, this is considered a Salam contract. The property owner must adhere to delivering the apartment at the specified time. If delivery is not possible, the buyer has no right but to reclaim their principal or wait for the apartment's delivery. They are not entitled to claim profits because the amount is a debt, and any increase on it is usury (riba).