Using the familiar spell metamagic feat you can have your familiar possess objects for you. This presents some interesting problems as familiars get a number of abilities relating to spellcasting that Possess Object seems intent on circumventing.
Target one object; see text
Duration 1 hour/level or until you return to your body
DESCRIPTION
This spell functions as magic jar, except you transfer your mind to a single object, animating it as if using animate objects, except your mind controls the object as if it were your own body. You cannot speak or cast spells while possessing the object. Because your original body is effectively dead while under the effect of this spell, this temporarily suspends disease, poisons, and other afflictions affecting you.
If you are successful, your life force occupies the host body, and the host's life force is imprisoned in the magic jar. You keep your Intelligence, Wisdom, Charisma, level, class, base attack bonus, base save bonuses, alignment, and mental abilities. The body retains its Strength, Dexterity, Constitution, hit points, natural abilities, and automatic abilities. A body with extra limbs does not allow you to make more attacks (or more advantageous two-weapon attacks) than normal. You can't choose to activate the body's extraordinary or supernatural abilities. The creature's spells and spell-like abilities do not stay with the body.
So is a Familiar able to access its Share Spells or Deliver Touch Spells abilities while inside a possessed object? It would give it access to some useful polymorph spells like Beast Shape and allow it to deliver persistent touch spells like Frostbite for some truly devastating attacks coming from a Gargantuan Animated Object's ability scores. It would also let it assume a useful size for traveling using Alter Self.
Alternatively, can you use a Greater Hat of Disguise while possessing an object to gain the benefits of Alter Self?
So is a Familiar able to access its Share Spells or Deliver Touch Spells abilities while inside a possessed object?
Yes. The Familiar isn't casting anything itself.
can you use a Greater Hat of Disguise while possessing an object to gain the benefits of Alter Self?
I guess that would depend on whether the possessed object can "wear" a hat.
I actually found the answer to the second question, pretty much all magic items that aren't automatic require a command word to activate whether they say so or not. Since you can't speak while possessing an object you can't use a hat.