Is a polymorphed familiar still considered a familiar for purposes of casting touch spells through them?
Finally, when you cast a spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.
All familiars, polymorphed or not, are merely conduits for your spells, and cannot cast spells themselves. Because you are not polymorphed, you can still cast a spell and deliver it through your familiar. The animals you can have as a familiar can't cast spells to begin with, anyway.
The CR limit ensures balance
Note that all of the creatures you can have as a familiar have CR0, except for the poisonous snake at CR1/8. This means that the number of creatures you can polymorph your familiar into is quite limited.
An explanation in the fiction
Your familiar is merely a spirit that takes on a "familiar form":
You gain the service of a familiar, a spirit that takes an animal form you choose...
Moreover, recasting the find familiar spell allows you to change its form:
If you cast this spell while you already have a familiar, you instead cause it to adopt a new form.
This suggests that the physical form of a familiar, which polymorph targets, is not as important as its connection to its master--after all, it is merely a spirit that can take on many mutable forms.
No. Polymorphed creatures can't cast spells
The effects of Polymorph over the target creature's new capabilities include the following (I have added emphasis to the relevant parts of the text):
The creature is limited in the actions it can perform by the nature of its new form, and it can't speak, cast Spells, or take any other action that requires hands or Speech.
The target's gear melds into the new form. The creature can't activate, use, wield, or otherwise benefit from any of its Equipment.
The familiar's ability to deliver one of your spells reads as follows:
Finally, when you Cast a Spell with a range of touch, your familiar can deliver the spell as if it had cast the spell.
A polymorphed creature cannot cast spells, and the ability to deliver a touch spell reads that the familiar is treated as the one casting the spell. Thus, it's not possible.