Currently (as of Requiem 2.0.0 beta 8), armor stands can not be possessed because they are not mob entities. However, they have attributes (including Health, etc), they can deal damage to and kill players with the use of the Thorns enchantment, along with several other characteristics of actual mobs.

According to the Minecraft Wiki, an Armor Stand has the following NBT:

• Tags common to all entities (such as Air, Fire, etc)
• Tags common to all mobs, except LeftHanded, DeathLootTable, DeathLootTableSeed, NoAI, Leash, CanPickUpLoot, and PersistenceRequired
• DisabledSlots
• Invisible
• Marker
• NoBasePlate
• Pose
• ShowArms
• Small

The Identity mod allows a player to disguise as an Armor Stand, which does work correctly and includes correct sounds.
The DisguiseLib mod allows an entity to disguise as an Armor Stand, which changes the appearance but not the sounds.

For Requiem interaction:

• Armor Stands are considered to be golems, allowing them to be freely possessed in a frictionless manner.
• If the Armor Stand has Marker:1b, it can't be possessed.
• Possible idea: If the Armor Stand has NoBasePlate:0b and a config option is set, it gets NoBasePlate:1b and a separate requiem:armor_stand_base entity is spawned which is just the base plate of the armor stand.
• Possible idea: The possessor of an Armor Stand may be able to open a GUI to change its Pose.
• Possible idea: If the possessed Armor Stand is holding 2 sticks, they can be used in order to set ShowArms:1b.

While I do understand that armor stands are technically living things, from a gameplay perspective they are just furniture... I feel like if you were able to possess an armor stand, you should also be able to possess paintings, and that's a lot less doable code-wise already.

That’s a great idea!

possessed armor style something would be pretty damn sick

Maybe you can only posses them if they're wearing a carved pumpkin/jack o' lantern.

Maybe make a new curse enchantment that when an entity or player is wearing it ethereal players can possess them even if they still have a soul! This is just a fun idea I wanted to throw out so no pressure to add it or anything!