Nope, unforunately not, you can’t get residency as Class5 driver since it’s still regarded as a non-skilled occupation… There has been several years of push from unions to get it listed as skilled occupation but nothing has happened. Still though, you can get work to residency -visa through few companies as long as they are accredited and you’re hourly rate is high enough. This mainly applies for tanker work, in general freight rates ares still a bit too low.
However, you can get residency through a relationship with kiwi 