No, your customer don’t need to buy license but keep in mind for each individual use you have to register the downloaded item. But for getting update features and support from theme author your client can purchase a license from themeforest.
The licenses you get for items you download and register from Envato Elements are only valid if you complete your End Products while your subscription is active. Then the license continues for the life of the End Products (even if your subscription ends).
But note that You have to be actively Subscribed to the platform(elements) when the project is going on and completed I.e you cannot stockpile items for future use and cancel the subscription.
All is fine but You have to be actively Subscribed to the platform(elements) when the project is going on and completed I.e you cannot stockpile items for future use and cancel the subscription.