The Order Domain cleric's Voice of Authority feature (GGtR, p. 26; TCoE, p. 31) says:
If you cast a spell with a spell slot of 1st level or higher and target an ally with the spell, that ally can use their reaction immediately after the spell to make one weapon attack against a creature of your choice that you can see.
If your ally, the intended target of healing word, is at 0 HP and making death saving throws, they're unconscious and presumably prone. Voice of Authority triggers after you cast healing word on them. When you cast healing word on them, they regain an amount of HP equal to 1d4 + your spellcasting ability modifier, which causes them to no longer be unconscious.
Since they are no longer unconscious, they can take actions and reactions as normal (i.e. actions on their turn, or reactions when their trigger occurs), assuming they're not affected by another condition or effect that prevents them from doing so. Thus, the ally you healed could indeed use their reaction to make a weapon attack against a creature of your choice that you can see, as long as the creature you choose is within range of the ally you healed.
As Miniman notes in a comment on Derek Stucki's answer, though, the attack would be made at disadvantage due to the character still being prone. (The character would continue being prone until at least their next turn; on their turn, they can use an amount of movement equal to half their speed to stand up.)
In addition, when the character was reduced to 0 HP, they would have fallen unconscious (which is also what causes them to fall prone). As a result, they would have dropped whatever they were holding, including a drawn weapon. The character would not yet be able to pick any such weapon back up (or draw a new one) after being healed – since they can only do so on their turn, using a free object interaction or the Use an Object action. Thus, the only attack they can make as part of this process is an unarmed strike or an attack with a natural weapon.