Can puberty cause thread veins?

Maybe. "thread veins" are much more common in women than in men, and there is probably a hormonal component to their incidence. Many women start to develop them shortly after maturation.
Veins and puberty. Spider (thread) veins get their name because they look like threads or spider legs. There are many reasons for their formation including genetics, hormones, sun exposure and injuries. Certainly, since there are many hormonal changes during puberty, the chances of spider veins forming exists. Usually spider veins forming at an early age are due more to genetics than hormones.
Yes. The rather sudden and significant rise in hormones, particularly estrogen, can be a cause for a few thread veins to develop.