Using Tasha's Cauldron of Everything Sidekick rules can you turn a familiar from the Find Familiar spell into a sidekick?
Whether or not a creature can become a sidekick is up to the DM
On page 4 of Tasha's Cauldron of Everything, the book details the content that will be covered in Chapter 4:
"Chapter 4 holds various rules that a DM may incorporate into a campaign."
Indicating that these are optional rules up to the DM's discretion.
We get more indication that it's up to the DM in this quote:
As DM, you determine whether there is sufficient trust established for the creature to join the group.
While this quote is questionably applicable to familiars, it further establishes that the DM is the one who chooses what can be a sidekick.
There is nothing that says you can't, but it comes with some limitations
There is no explicit statement saying that a creature from the Find Familiar spell cannot become a Sidekick. On page 142, however, the designers do specify limitations on what can become a sidekick:
A sidekick can be any type of creature with a stat block in the monster Manuel or another D&D book, but the challenge rating must be 1/2 or lower.
Which a familiar certainly falls under. The book goes on to say this:
You can take that stat block and add to it
Indicating when you create a Sidekick, it does not lose anything.
Each sidekick class states what is necessary to qualify for the class.
The Expert says:
To gain the Expert class, a creature must have at least one language in it's stat block that it can speak.
The Spellcaster possesses this exact restriction as well. The Warrior, however, has no such restriction, and the book even offers an illustration of a Wolf Warrior. However, the book provides an illustration of a Kenku expert, which cannot speak a language itself in its monster stat block, only understand them. The Raven familiar, similarly, can mimic sounds, it cannot speak any language. The Familiars gained by a Pact of the Chain Warlock, however, can often speak languages.
Familiars are also unable to take the attack action:
A familiar can't Attack, but it can take other Actions as normal.
And since we are adding to the creature's capabilities, it could easily be interpreted that it will keep this restriction. So having a familiar that is a Warrior would only serve to improve its defensive abilities.