Whether this proves or disproves the theory is up to you: We do not know of an account of an Animagus performing magic (in animal form), so we do not know if they can.
Rita Skeeter transforms into a bug, and bugs the hell out of people by bugging them... But at no point is she performing any inherent feats of magic; she probably merely remembers all of the info she hears, and then writes it down once back in human form. She also cannot escape from Hermione's jar trap; she (Rita) would need to know how to Apparate which I'm not sure if she can, but still...
Sirius (in Animagus form) fights Lupin (in Werewolf form). They are clearly bare-fisted(toothed) fighting; could Sirius have stunned Lupin? maybe, but he doesn't. Also, I seem to recall Padfoot opening a door with his paws (or was that the Movie version?).
Wormtail is in his Animagus form (Rat) for many years - almost the entire duration from Voldemort's first demise and all the way until the events of Prisoner of Azkaban. He does not seem to perform any inherent feats of magic.
Babbity Rabbity is a character from the Tales of Beedle the Bard who apparently transforms into a Rabbit; she doesn't actually perform any magic (in her Animagus form), but she does threaten to do so (Cruciatus curse?).
Also, from the Tales of Beedle the Bard, Albus Dumbledore states:
Animagi do not retain the power of human speech while in their animal form, although they keep all their human thinking and reasoning-powers.
-Albus Dumbledore (footnotes), Tales of Beedle the Bard (Babbity Rabbity and Her Cackling Stump)
Human 'thinking' and 'reasoning-powers' could mean their ability to perform magic, but all we know is that the only thing they lose is their 'human speech'.