Your assumption is correct. Veinard is a colloquial word (but not vulgar). It means "someone who is very lucky".
According to the Larousse definition:
Veinard : Qui a beaucoup de chance.
Using the word "petit" before it gives it a petty meaning, as if the person writing the sentence was expressing a form of jealousy towards these newly rich people.
Origin of the expression
This adjective is taken from Avoir de la veine. The origin of this expression is not officially determined. However, the most popular (and logical) explanation is that it would come from the age of gold diggers.
Indeed, a veine expresses a part of a mine where the mineral is located.
À force de creuser, il est tombé sur une veine d'or.
Since finding gold is a lucky thing, it possibly derived to the expression we know today.