Hi Pris, if you intend to use the green carts with your JR Pass, you are obligated to reserve seating beforehand, yes.
You are however allowed on any train without booking with your green pass, but will have to use the normal cars on the train instead.
Seat booking is free of charge and can be done up to 30 days beforehand. So if you have your route planned, feel free to use the kiosks or talk to any JR staff to have the seats booked free of charge.
Let me know if you have any other questions and have a wonderful stay in Japan! 🙂
With kind regards, Jonathan H.