This is a tricky question, but most likely, no.
(However see the"yes" possibility at the end! )
Here are the relevant sections:
Only a normal, unmodified animal may become a familiar.
In this context, "normal, unmodified" means exactly as the start block entry details the creature in question. All subsequent changes are a mathematical function of the summon familiar class feature. As the class feature is not natural HD progression, no feats are gained.
Had the class feature said:
Only a normal, unmodified animal may BE a familiar.
...then our answer would be done. However, because of the ambiguous wording, we are left with the question of: "Can a familiar gain xp and eventually HD (and thus feats) outside of and in addition to, just being a familiar?"
If we peruse every stat block of every character possessing a familiar in every book, article, and product published by WotC, we will find that in virtually every instance, a familiar does NOT have any feats not listed in the standard stat block for the base creatures, even when the accompanying pc or npc is of high level.
However, while strongly indicative, that is circumstantial. Yet there is one more source we can turn to extract an rules-as-intended example.
The psicrystal, unlike other base creatures, normally starts off as an unliving chunk of crystal, animated by a fragment of the owners personality. It only exists if summoned into existence by the class feature. It also has a monster entry. This monster listing has an entry for advancement, which is:
"—" has a special rules meaning in the start block. It means "non-ability". When a creature has a non-ability, it is not the same as having a zero in that ability, it means that the ability does not exist, it can't make saves (if a stat) or checks (if a characteristic). The ghost is another example of a creature with a non-ability.
As such, the ability to advance does not exist for the psicrystal. It can't gain HD as a construct, and thus it can never gain feats.
While this is, again, circumstantial evidence, this seems to show the intended interpretation of the rules for familiars and familiar-like creatures. That the familiar cannot gain feats, so long as it is a familiar.
For myself, I think it would be interesting for there to be an "emancipated familiar" prestige, similar to the "emancipated spawn" prestige, for familiars that survive the death of their master. Which would ideally have an entry for ex-familiars becoming a familiar again. (And would presumably have gained a few HD in the interim).
Obviously, for familiars whose base creature DO have feats, there are various methods to swap out existing feats, or even adding feats. The dark chaos trick, psychic reformation, magic items, a few spells and powers, and special locations can affect feats in various fashions.
Thus, to answer your question the character in question will need to select as a familiar a creature that already has feats. Enough feats for their intended result.
The yes possibility.
It is worth mentioning that there is an alternative thought process with a different result that I recently was made aware of.
The term HD is defined and feats, skills, ability increases, hit point increases and so forth are all based on HD.
Therefore, one can assume that all those features are a function of Hit Dice. Thus any feat or class feature which grants a companion that also assigns an increase in hit dice of any kind should therefore mean that the creature in question also automatically gains everything that goes along with a hit die.
With this line of thinking, then psicrystals can gain feats when they gain enough hit dice; a circumstance of which I am happy about, as I've long felt that such creatures should be able to do so.
However, in the case of a psicrystal, this line of thinking ignores the explicit non-ability regarding advancement in the psicrystal entry, though it should work for other creatures in the companion slot.