Mega Man 2 was composed by Takashi Tateishi. He also composed the music for Willow and LED Storm for the arcades and Side Arms Special on the PC Engine. Sadly, he retired from composing after leaving Capcom in 1990.
Mega Man 3 was composed (mostly) by Yasuaki Fujita. He also worked on Darkwing Duck (NES), The Little Mermaid (NES), Tenchi wo Kurau II (NES), Who Framed Roger Rabbit (GB), Final Fight 2 (SNES), Breath of Fire (SNES) and a few other non-descript mahjong and card games for the Super Famicom. Also composed Solar Man's theme for the somewhat recent Mega Man 10.