If you can cast Warding Bond on yourself, via touch, then Yes, your familiar can do it.
If you can't cast Warding Bond on yourself, via touch, then No, your familiar cannot do it.
The embedded question required to answer the question posed is:
"Can you cast Warding Bond on yourself?"
Miniman's answer suggests that you can.
The following RAW-based analysis suggests "No, You Can't" based on the specifics in the spell description rather than the general rules cited in the "Yes" answer. The spell relies on there being two creatures (plural) for this magical bond to form and protect one of them. From the spell description:
The spell ends if you drop to 0 hit points or if you and the target become separated by more than 60 feet. It also ends if the spell is cast again on either of the connected creatures.
A tweet by Jeremy Crawford suggests that you can't:
Sean @Lord_Sicarious · 6 Oct 2016
@JeremyECrawford can you cast Warding Bond on yourself? Its rules seem to assume two people, but no explicit limitation.
Jeremy Crawford @JeremyECrawford · 6 Oct 2016
Warding bond—you can't target yourself with it. #DnD
"Specific over General" supports No over Yes.
Each DM will rule on what makes most sense to that DM. Since the impact isn't game breaking, it probably doesn't matter. A "Yes" ruling provides a one-hour period where a cleric gets +1 AC and +1 to saves for the price of a level 2 spell after touching him / her self while wearing two platinum rings. A "No" ruling means the cleric uses the 2d level slot on something else.
Analysis of the Spell to Support the No Answer (Specific over General)
Casting Time: 1 action
Components: V, S, M (a pairof platinum rings worth at least 50 gp each, which you and the target must wear for the duration)
The italicized text implies two creatures -- a caster and a target -- each wearing a platinum ring, but it does not state that explicitly in this part of the spell description.
This spell wards a willing creature you touch and creates a mystic connection between you and the target until the spell ends.
"A willing creature" isn't necessarily only "another willing creature," so one can argue that the caster is "a willing creature" who can touch his/her self. That said, the second half of the sentence implies two parties being involved: the caster and a target. this spell
... creates a mystic connection between you and the target until the spell ends.
- "You create a mystic connection with yourself"
is not the same statement as
- "you create a mystic connection between you and (any other different creature than you.)"
The "general rule" arguments about spell targets, self spells, and touch spells seems to have raised its head again. (See a previous discussion on Paladin Smite spells).
Two parties being involved is the common sense / common usage reading of this spell description.
While the target is within 60 feet of you, it gains a +1 bonus to AC and saving throws, and it has resistance to all damage.
We see a second and a third person usage, implying the presence of another creature other than the caster.
Also, each time it takes damage, you take the same amount of damage.
"You take damage" and "it" takes damage: second person and third person references. Two different persons, to different creatures are damaged.
The spell ends if you drop to 0 hit points or if you and the target become separated by more than 60 feet.
The caster reaching a zero-hit-point condition ending a spell is common result. Neither a pro nor con element for this question. Outside edge case magical effects, you can't be separated from yourself so this makes no sense in a case other than two creatures being involved.
It also ends if the spell is cast again on either of the connected creatures. You can also dismiss the spell as an action.
Creatures, plural. Two creatures, a caster and a target who needs to be touched while wearing the appropriate platinum ring.
What was implied in previous language is specified at the end of the spell description. "Either" obviously refers to more than one party/creature.
By reading the specifics of the spell description, the spell requires two creatures, each wearing the appropriate platinum ring, one touching the other, and both staying within 60' of each other for up to an hour for the spell to provide the damage reduction to recipient of this spell, as well as the AC and Saving throw bonuses.
No, you can't cast this spell on yourself because it requires two creatures, each wearing that platinum ring, to create the spell effect.