After Izuku was considered quirkless he looked towards his dad who is the head leader of the Mafia. After getting treated like shit by his school he decided to fight back as he used a weapon and used it against his main tormentor Katsuki Bakugo as he left him wounded as everyone was totally afraid of him. When the principal told Inko about this she didn't care as she was tired of the bullshit as the phone was in as her husband came pissed off.
Family + wife Inko Midoriya (mother)
Hisashi Midoriya (father)
Momo Yaoyorozu-Midoriya(wife and part of the Yaoyorozu Mafia)
Eri Midoriya (daughter)
Yes Izuku "was" considered quirkless but it was due to Hisashi taken his quirk which is AFO 2.0