There is no single "American Girl Online Game" — the brand offers several digital experiences:
Free Website Games
americangirl.com/play: Mini-games (dress-up, pet care, quizzes, simple arcade). No account needed. Works desktop/mobile browser.
Mobile App (Free + IAP)
American Girl World (iOS/Android): Avatar creator, mini-games, stories, social hub. Free download; subscriptions ($4.99/mo or $29.99/yr) unlock all content.
Retired / Legacy
Innerstar University (2010–2015): 3D virtual world for doll owners (code from book). Shut down.
Doll-Specific Sites (e.g., Girl of the Year minisites): Most retired; some archived on Internet Archive.
Requirements
Website: Browser + internet.
App: iOS 14+ / Android 9+; Apple/Google account for IAP.