There is no web API. You’ll have to be connected to IRC and request message tags which will contain the user’s color with each of their messages.
CAP REQ :twitch.tv/tags
@color=#0000FF;emotes=16156:0-7;subscriber=1;turbo=0;user_type=mod :joe_user!joe_user@joe_user.tmi.twitch.tv PRIVMSG #somerandomchannel:lirikMLG is awesome!