The two surfers eliminated Elin Tawharu, one of the pre-event favourites in the women's field.
Other surfers to make it into the final eight included 2019 New Zealand Surf Series winner Raiha Ensor and New Zealand junior representatives Ava Henderson and Natasha Gouldsbury.
Stairmand eased through his heat to advance to the open men's quarterfinals alongside Tom Robinson, from Whangarei. Raglan's Caleb Cutmore had the highest score in the heats, 13.37.
Taylor O'Leary, Navryn Malone and Jayden Willoughby were others to advance with impressive performances.
Vette started the champs with a win in his open men's heat. His combined score of 8.16 from his two best waves was enough to beat Aucklanders Cassidy Mann and Jackson Peak and Dunedin's Ruben Peyroux.
The National Surfing Championships is the first of several domestic events where Kiwi surfers will get the chance to push for selection for the New Zealand team to attend the World Surfing Games in May.
The event finishes on Saturday.