Is it permissible for visitors to the Sacred Mosque (Al-Haram Al-Makki Al-Sharif) to ask Facebook users to write their supplications for them, and should people be advised to stop this act?

There is no harm for a Muslim to ask his brother if he has a need for which he would like him to supplicate, as the Prophet, peace and blessings be upon him, did with Rabī‘ah ibn Ka‘b al-Aslamī.

However, one should beware of corrupt intentions, such as the intention to show off one's piety. Rather, one should intend, by supplicating for his brothers, to draw closer to Allah and benefit himself.

Moderation in this is also necessary, so one should not spend all his time supplicating for people and neglect other acts of worship.