Short answer? Yes.
Long answer?
First we must ask ourselves: Does using Distant Spell on a spell and casting it occur at the same time? Or would the metamagic change the range beforehand and disqualify it from being able to be cast from your familiar?
As per Distant Spell (PHB 102)
When you cast a spell that has a range of touch, you can spend 1
sorcery point to make the range of the spell 30 feet.
As per Find Familiar (PHB 240)
Finally, when you cast a spell with a range of touch, your
familiar can deliver the spell as if it had cast the spell. Your
familiar must be within 100 feet of you, and it must use its reaction
to deliver the spell when you cast it.
These both state "When you cast a spell" indicating that they would occur at the same time. since they occur simultaneously, the Distant Spell Metamagic would not disqualify it from being cast from your familiar. This is further supported in Do the rules for combining magical effects apply to non-spell effects? where it is confirmed that multiple effects can exist on a target at the same time. The target in this case being the touch spell that you want to extend the range of.
Now that we've established that it works, We must consider how it works. So to answer your question, here are my thoughts:
Assuming you are within 100 ft of your familiar, and your spell target is within 30 feet of your familiar, there are several situations to consider when answering this:
Situation 1: The spell target is unobstructed to both you and your familiar.
This combo should work as expected.
Example 1: A goblin is in an open field, but your familiar is nearby the goblin. You use your action and expend 1 sorcery point and cast a touch spell on the goblin. The goblin is deeply concerned by this turn of events.
Situation 2: The spell target is only unobstructed to you.
Find Familiar treats the touch spell as though it was cast from your familiar. (See above) as per A Clear Path To The Target (PHB 204) "To target something, you must have a clear path to it, so it can't be behind total cover."
Example 2: A goblin is down a hallway. Locked in the next room beside it is your familiar. Since the goblin has full cover you are unable to use your combo.
Situation 3: The spell target is only unobstructed to your familiar.
Since your familiar is treated as the origin point of your spell, it might work! Under Unseen Attackers and Targets (PHB 194-195)
When you attack a target that you can't see, you have disadvantage
on the attack roll. This is true whether you're guessing the
target's location or you're targeting a creature you can hear but not
see. If the target isn't in the location you targeted, you
automatically miss...
Even though your familiar can see the spell target, you cannot, so this is going to be tougher to pull off.
Example 3: A goblin has positioned itself behind a large rock, giving itself full cover from you. Your familiar has positioned itself on top of said rock. You make an educated guess at where the goblin is and use your action and expend 1 sorcery point and attempt to cast a touch spell on the goblin. The DM will tell you if you succeed.
Situation 4: The spell target is unobstructed to neither you nor your familiar.
There is no path for the magic to take from either point. You are unable to meet the criteria needed to cast the spell. You are unable to use your combo.
Example 4: You, your familiar, and a goblin are all locked in separate rooms. Nobody here has an unobstructed path. The spell is impossible to cast.
Further thoughts:
This is a clever way to synergize niche bits of mechanics and should be lauded. Additionally, this tactic isn't free, as the character will need to expend both spell slots and sorcery points to pull it off, making it lore-friendly as a more advanced/difficult and specialized casting style.
