How bad is the karma of feeding meat to your children?

If you can introduce her to vegetarianism and at least not encourage meat consumption, it would be of benefit to you both.

You do not specify if you yourself are vegetarian or vegan. I bring this up because, first and foremost, the precepts are yours to keep, and not to make you feel that you must force them on others. However, if you are vegetarian or vegan, your example in the family, who are at least omnivores, as you say, can serve to educate the young child that there are concerns about eating animal flesh. These concerns are that it is a cause of suffering for the animals, is considered to be a health risk for humans, and is implicated in the production of the majority of global-warming chemicals that are contributing to the collapse of the life-supporting ability of the Earth.

As to the specific teachings found within Buddhism, I find the Buddha’s own words to be authoritative:

Why is this important?

In explaining to you the rules of the Vinaya, I have frequently emphasized three good lessons, namely, (1) the only way to keep the Precepts is first to be able to concentrate the mind; (2) by keeping the Precepts you will be able to attain Samadhi; (3) by means of Samadhi one develops intelligence and wisdom. Having learned these three good lessons, one has gained freedom from the intoxicants and hindrances.

Ananda, why is concentration of mind necessary before one can keep the Precepts? And why is it necessary to keep the Precepts before one can rightly practice dhyana and attain Samadhi? And why is the attainment of Samadhi necessary before one may attain true intelligence and wisdom? Let me explain this to you. All sentient beings in all the six realms of existence are susceptible to temptations and allurements. As they yield to these temptations and allurements, they fall into and become fast bound to the recurring cycles of deaths and rebirths.

Regarding Violence to Animals:

The next important hindrance and allurement is the tendency of all sentient beings of all the six realms of existence to gratify their pride of egoism. To gain this one is prone to be unkind, to be unjust and cruel, to other sentient beings. This tendency lures them into the bondage of deaths and rebirth, but if this tendency can be controlled they will no longer be lured into this bondage for right control of mind will enable them to keep the Precept of kindness to all animate life. The reason for practicing dhyana and seeking to attain Samadhi is to escape from the suffering of life, but in seeking to escape from suffering ourselves, why should we inflict it upon others? Unless you can so control your minds that even the thought of brutal unkindness and killing is abhorrent, you will never be able to escape from the bondage of the world's life. No matter how keen you may be mentally, no matter how much you may be able to practice dhyana, no matter to how high a degree of Samadhi you may attain, unless you have wholly annihilated all tendency to unkindness toward others, you will ultimately fall into the realms of existence where the evil ghosts dwell.

There are three ranks of these ghosts: the highest are the mighty ghosts, the next are the Yaksha ghosts who fly in the air, and the lowest are the Raksha ghosts that live under the earth. Each of these ghosts has his double that disguises itself as having attained enlightenment. After my Parinirvana in the last kalpa these different kinds of ghosts will be encountered everywhere deceiving people and teaching them that they can eat meat and still attain enlightenment. But how can any faithful follower of the Lord Tathagata kill sentient life and eat the flesh?

You of this great assembly ought to appreciate that those human beings who might become enlightened and attain Samadhi, because of eating meat, can only hope to attain the rank of a great Raksha and until the end of their enjoyment of it must sink into the never ceasing round of deaths and rebirths. They are not true disciples of Buddha. If they kill sentient beings and eat the flesh, they will not be able to escape from this triple world. Therefore, Ananda, next to teaching the people of the last kalpa to put away all sexual lust, you must teach them to put an end to all killing and brutal cruelty.

If one is trying to practice dhyana and is still eating meat, he would be like a man closing his ears and shouting loudly and then asserting that he heard nothing. The more one conceals things, the more apparent they become. Pure and earnest bhikshus and Bodhisattva-Mahasattvas, when walking a narrow path, will never so much as tread on the growing grass beside the path. How can a bhikshu, who hopes to become a deliverer of others, himself be living on the flesh of other sentient beings?

Pure and earnest bhikshus, if they are true and sincere, will never wear clothing made of silk, nor wear boots made of leather because it involves the taking of life. Neither will they indulge in eating milk or cheese because thereby they are depriving the young animals of that which rightly belongs to them. It is only such true and sincere bhikshus who have repaid their karmic debts of previous lives, who will attain true emancipation, and who will no more be bound to wander to this triple world. To wear anything, or partake of anything for self-comfort, deceiving one's self as to the suffering it causes others or other sentient life, is to set up an affinity with that lower life which will draw them toward it. So all bhikshus must be very careful to live in all sincerity, refraining from even the appearance of unkindness to other life. It is such true hearted bhikshus who will attain a true emancipation. Even in one's speech and especially in one's teaching, one must practice kindness for no teaching that is unkind can be the true teaching of Buddha. Unkindness is the murderer of the life of Wisdom. This is the second admonition of the Lord Buddha as to the keeping of the Precepts.

(Taken from Surangama Sutra, Chapter Two, Importance of Keeping the Precepts, found at: https://medium.com/p/e84c1abb029b )

And just a final, personal note: I find the argument that it is ok to eat animal flesh so long as you have no involvement in the killing of the animal to be a feint-hearted attempt to escape one’s culpability in providing a market for such murdered flesh. Without a willing buyer, there would be no killing.

If I raise this kid not vegetarian, will that be bad karma? Her mother is an omnivore and the kid is being raised in an omnivore household. It would be very difficult to change her diet. Is it my reponsibility to make her vegetarian?

The Buddha's advice about this matter in MN 55:

Then Jīvaka went up to the Buddha, bowed, sat down to one side, and said to the Buddha:

“Sir, I have heard this: ‘They slaughter living creatures specially for the ascetic Gotama. The ascetic Gotama knowingly eats meat prepared on purpose for him: this is a deed he caused.’ I trust that those who say this repeat what the Buddha has said, and do not misrepresent him with an untruth? Is their explanation in line with the teaching? Are there any legitimate grounds for rebuke and criticism?”

“Jīvaka, those who say this do not repeat what I have said. They misrepresent me with what is false and untrue.

In three cases I say that meat may not be eaten: it’s seen, heard, or suspected. These are three cases in which meat may not be eaten.

In three cases I say that meat may be eaten: it’s not seen, heard, or suspected. These are three cases in which meat may be eaten.

Please read "Why is contributing to the market demand for meat not wrong?".

According to most schools of Buddhism, it is against the first precept to kill an animal yourself, but it is ok to buy frozen meat from the supermarket for consumption.

Let's say you go to a restaurant as a customer. If the restaurant prepares your meal order using frozen meat, then that's ok. But if you have to select the animal for slaughter (which happens in some Asian countries, for seafood dishes for e.g. you need to pick your lobster from an aquarium), then that breaks the first precept.

As long as you did not do the killing yourself or select the animal and ordered the butcher or chef to slaughter for you, it is not a violation of the first precept.

To summarize from that question:

• Avoid killing animals and avoid becoming the direct cause of the killing of animals
• Avoid having livelihood on the business of meat
• Avoid consuming meat that is from an animal that is seen, heard or suspected to have been slaughtered specifically for you
• You can purchase and consume frozen meat from the supermarket (that was already dead long before you arrived at the market)
• You can order a meal from a restaurant, which is based on frozen meat

Why? This is because you did not have the intention to kill that animal. You are simply buying meat that was no longer alive when you first encountered it.