Yes. You can use a wooden board and set at a chess tournament. The key is to make sure that the dimensions of the board and pieces are correct and that the contrast is also appropriate.
The player with the black pieces gets his choice of equipment and what side the clock is on.
I hope this helps.
I'm playing in my first tournament this weekend, and I'm currently looking to buy a tournament chess set. A couple years ago I bought a nice wooden set (10cm king, 4cm diameter, triple weighted), so I was thinking of using the wooden pieces and then buying a wooden board to go along with it. But, is using a wooden set at amateurish tournaments considered a faux pas? Or does everybody just use the vinyl/plastic sets?