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.
Depends on the type of poison. If it's an injury poison, unless the wielder has a magical weapon that CAN pierce the skin, then, no. But if it's a CONTACT poison, well, looking at their immunities, they are NOT immune to contact poisons. So if I had a non-magical blade loaded up with a contact poison, just touching them would be enough to get the effect of the poison.
Types of poison in D&D 5 include, with my notes in italics:
Contact: Contact poison can be smeared on an object and remains potent
until it is touched or washed off. A creature that touches contact
poison with exposed skin suffers its effects. Therefore they suffer the effects if they are touched, no need for injury.
Ingested: A creature must swallow an entire dose of ingested poison to
suffer its effects. The dose can be delivered in food or a liquid. You
may decide that a partial dose has a reduced effect, such as allowing
advantage on the saving throw or dealing only half damage on a failed
save. Hard to do in combat.
Inhaled: These poisons are powders or gases that take effect when
inhaled. Blowing the powder or releasing the gas subjects creatures in
a 5-foot cube to its effect. The resulting cloud dissipates
immediately afterward. Holding one’s breath is ineffective against
inhaled poisons, as they affect nasal membranes, tear ducts, and other
parts of the body. Very effective against them, yes.
Injury: Injury poison can be applied to Weapons, ammunition, trap
Components, and other Objects that deal piercing or slashing damage
and remains potent until delivered through a wound or washed off. A
creature that takes piercing or slashing damage from an object coated
with the poison is exposed to its effects. Therefore, if no damage is taken, the poison cannot be delivered.
EDIT: However, in the comments, it's been pointed out that the lycanthrope immunity has less to do with not being able to pierce the skin at all and more to do with the fact that any non-magic/non-silver wound damage from those particular sources may close pretty much instantly. A DM could rule this way, but mechanics-wise, according to the rules, an injury poison would not work because it specifically states "a creature that takes piercing or slashing damage" and lycanthropes DO NOT take piercing or slashing damage from non-magic, non-silver weapons.
Best Answer
The target is poisoned
The target is poisoned and suffers additional effects as long as they remain poisoned. A similar wording exists for many spells that inflict the Charmed condition and some additional effect, for example Dominate Person.