I have a temporary residence card with the notation "STUDENT." Do I have access to the labor market?

YES. The annotation "student" is placed on the residence card in the case of a permit granted for studies, i.e., under Article 144 of the Law on Foreigners.
A foreigner with a temporary residence permit granted in the Republic of Poland due to the circumstances referred to in Article 144 of the Law on Foreigners (due to full-time studies in Poland) is exempted from the obligation to have a work permit.