You will still be able to reserve seats, but you should make sure you reserve them well in advance: 3-4 days minimum. Golden week trains tend to be busy, so you should give yourself plenty of time to make a reservation. If you're travelling from Tokyo, then you can reserve some of your seats using the JR-East online seat reservation system. You'll need to enter credit card details, but your card will only be charged if you fail to pick up the reservation tickets by 9pm the day before at the latest.
Alternatively, if you want to book once you arrive in Japan, any JR ticket office will be able to arrange seat reservations for any other train in Japan at any other date (within reason), so you don't have to make the reservation at your departing station.
If you plan ahead a little you should be fine, and travelling with the rest of the Japanese is a great experience!
Have a great time in Japan!