Is it permissible to buy a digital camera for one of the brothers, and is the buyer sinful if the recipient uses it for an impermissible act?

There is no blame upon you for purchasing the camera for the one who requested it, unless you predominantly believe he will use it for something unlawful. If he later uses it for something unlawful, there is no sin upon you, for Allah Almighty says: "And no bearer of burdens will bear the burden of another." However, if you predominantly believe he will use it for something unlawful, then it is not permissible for you to buy it for him, because that falls under cooperating in sin, for Allah Almighty says: "And cooperate in righteousness and piety, but do not cooperate in sin and aggression."