Is it permissible for my uncle to take money in exchange for vacating an apartment he does not live in, at the request and with the consent of the apartment's owner who wishes to sell it?

Your uncle has no right to the apartment's price. What the owner mentioned might be an inducement to vacate the apartment. The rental system that perpetuates the lease is Islamically invalid because the duration must be known in the lease contract, without dispute. Therefore, your uncle should hand over the property to its owner and not demand anything from its price. It is better for him not to accept anything from the owner.