As far as I am aware, there is no way to give a creature the incapacitated condition directly. However, there are 4 conditions that give a creature the incapacitated condition as part of their effects. These are:
- Paralysed
- Petrified
- Stunned
- Unconscious
There are a number of ways to apply these conditions without damaging your beloved pet. Within the Warlock spell list, the ones I could find are:
- Hypnotic Pattern
- Hold Monster
- Eyebite
- Flesh to Stone
- Power Word Stun
- Imprisonment
However, only Flesh to Stone, Imprisonment and Power Word Stun last longer than 1 minute. Flesh to Stone is severely not recommended, since you really shouldn't be able to harvest posion from a statue. Imprisonment is expensive, but will work well for this task. Power Word Stun is probably the best option, since your creature can choose to fail its saving throws till you're finished.
On the other hand, if you have an allied Bard, Cleric, or Wizard, the 3rd-level spell Feign Death suits your needs perfectly. It will give a willing creature the incapacitated condition for an hour. A Cleric wouldn't even need to waste a spell learning it, since Clerics know all their spells automatically. Since it's a ritual, he wouldn't even need to waste a spell slot on casting it! All you have to do is talk him into preparing it.
Of course, all this is moot if your DM is willing to allow you to extract poison simply on the basis that your wyvern is tame and willing to allow you to do what you want to it. Given that he's giving you a CR 6 creature for a mount, a little bit of poison is really not a big deal.
No, if these attacks were meant to inflict the poisoned condition they would specify. For example, the Sprite has an attack that does this:
Shortbow.
Ranged Weapon Attack:
+6 to hit, range 40/160
ft., one target.
Hit:
1 piercing damage, and the target must
succeed on a DC 10 Constitution saving throw or become
poisoned for 1 minute. If its saving throw result is 5 or lower,
the poisoned target falls unconscious for the same duration,
or until it takes damage or another creature takes an action to
shake it awake.
The Giant Scorpion and the Poisonous Snake don't say anything about the poisoned condition, so they don't inflict it.
Best Answer
RAW, your familiar must be incapacitated:
Since your familiar can’t die, and you can’t reduce its hit points to 0, we’ve got to find another solution.
Fortunately, there is an easy one: your familiar still needs to sleep.
From the sleeping rules listed in Xanathar’s Guide to Everything, we see:
And from the Player’s Handbook,
So wait until your little buddy is asleep, and proceed as described in the Dungeon Master’s Guide.
Alternatively, a generous DM could just rule that since you can communicate telepathically with your familiar and they are friendly to you, you can just do it. But there is a way to do it RAW with little trouble.1
1You should still discuss this with your DM. The question remains of what happens to harvested poison when your familiar disappears or changes form. A similar problem is addressed in this Q&A concerning harvesting materials from a wild shaped Druid. Differing answers on that question were well received, and it seems there is no RAW answer, so this is probably best left up to your DM.