Typically only creatures that get an alignment subtype can have natural attacks be treated as that alignment for purposes of damage reduction. PCs don't bypass Dr based on their alignment either.
Amulet of mighty fists is one way to do it. You can go for the plusses or take an alignment based enhancement on it.
Firstly, animals tend to just be neutral...so there is that.
Anyway, DR/alignment doesn't mean that your DR is cut through simply by creatures of that alignment, but by attacks that are literally infused with the energy of that alignment. In this game, alignments like good and evil are as part of the elements as much fire or water for these kinds of things. There are a few ways to get this type of infusion.
One of the first ways is to be a creature of that alignment's plane. Demons, for example, are creatures born in the Abyss, which is literally a plane of chaos and evil. Being made of that plane's substance means that it is literally infused with chaos and evil, and can harm creatures vulnerable to either of those. Unfortunately, there are only a couple of ways to turn your animal companions into such outsiders (aasimar have a racial feat for it).
A more practical way is to enchant your attacks so they count as that alignment. Holy, unholy, chaotic, and axiomatic are all weapon enhancement that can be applied to weapons (and an amulet of mighty fists, which is the typical way to enhance natural attacks) and they all get through the appropriate DR and deals additional damage to creatures of the opposite alignment (which are the only ones to have that kind of DR anyway). But unforunately, each of these cost +2 enhancement bonus... Oh, you can also ignore all alignment DR if your enhancement bonus (the straight +'s to attack and damage) is +5...which is not practical for AoMF due to price...
There are also various spells and abilities that can also have weapons treated as an alignment. For example, the spell "Bless weapon"....which doesn't work on natural attacks (only manufactured weapons). Yeah, that is a serious problem for you: the balancing factor for having animal companions is that sometimes they are simply ineffective since they can only use natural attacks, which have these kinds of restrictions.