Pretty sure the site does not.
OAuth2

You are able to login with chess.com credential sin some apps, for example the SquareOff app allows you to login with Chess.com details to use their boards here. Not sure what protocol that uses, but some form of external authentication must be available.

There are hints that an interactive API for chess.com exists, like lichess.org has. Since Squareoff, DGT and others interact with chess.com, it is likely that it exists, but it's not public. I would contact support with exactly what you want to do, and see if they can hook you up.
Hello devs,
do you know if chess.com has own OAuth2 system? I would like to add button and action like "log in with chess.com" to my project.
Thanks for answers.