The Bible has never clearly commanded vegetarianism and eating meat is certainly permitted in New Testament times. We know that Jesus, when He appeared to his disciples after His resurrection, ate fish (Luke 24:41 - 43).
We also know that, under the Law of Moses, an enormous number of animals were ritualistically slaughtered and offered to God by various burnt sacrifices (Leviticus 1, 3 - 7). The priests were allowed to eat the meat from what they offered under certain limitations (Leviticus 7:31 - 35).
It is not a sin to kill an animal and then eat it (provided it was considered clean). God had his priests at the temple kill thousands of animals over the centuries. They represented the sacrifice of Jesus for humanity's sins in advance symbolically.
Permission to Eat Meat
Perhaps the best example of God's direct permission to eat meat comes from celebrating the Passover. Each Israelite family was to take a lamb or goat, kill it, and (in the first Passover before leaving Egypt) put its blood on the doorposts and lintel. They were then to eat the lamb by morning and burn the leftovers before daybreak. This is described in Exodus 12. Many centuries later, John the Baptist proclaimed that Jesus was the Lamb of God offered to take away the world's sins (John 1:29, 36).
Abraham and Noah had to kill animals in order to make sacrifices to God, which was long before the time of Moses and the establishment of the Levitical priesthood (Genesis 8:20; 22:7 - 8, 13).
The Law Permits It
Under the law of clean and unclean foods, Israelites were allowed to eat the meat of cows, sheep, and goats, but weren't allowed to ingest pigs, camels, and rabbits (Numbers 11, Deuteronomy 14:3 - 21). These general rules are the greatest proof God does not require vegetarianism of his people.
Although God did not like how Israel in the wilderness demanded flesh to eat when they had a bad attitude, and punished them for it, He was still willing to provide them with a huge number of quail to eat in one special situation (Numbers 11:4, 13, 31 - 33). Obviously, it cannot be said that eating meat was deliberately sinful, or God wouldn't have sent them the quail.
A Counter Argument
There is one interesting counter argument worth examining. The argument is that Adam and Eve, when they were created, were not directly told they could eat the meat of animals. Although this may be true, the Bible does state that God gave man authority over the animals and was told to rule over them.
And God blessed them. And God said to them, "Be fruitful and multiply, and replenish the earth, and subdue it; and have dominion over the fish of the sea and over the fowl of heaven and over every living thing that moves upon the earth." (Genesis 1:28, HBFV).
There is also no doubt that God, after Noah's flood, gave humanity permission to eat meat (after the blood is drained away.
Every moving thing that lives shall be food for you, even as the green herb I have given you all things. But you shall not eat of flesh with the life in it - which is its blood. (Genesis 9:3 - 4, HBFV).
Future Food?
The Bible teaches that animals will not eat each other during the Millennium (Isaiah 11:6 - 9). That said, since there will animal sacrifices during this period (Ezekiel 40:39; 42:13; 45:15-25), it seems likely humans will continue to be allowed to consume the flesh of beasts.
It should be stated that we should not abuse animals even though God has given us authority over them and allows us to eat their meat. Balaam was criticized by his (miraculously) talking donkey and by the Angel (Messenger) for beating the animal when it saw the Angel of God with a drawn sword in front of him and refused to move (Numbers 22:22 - 35).
In conclusion, the Bible does teach that humans can eat meat if they so wish.
When the Lord your God shall enlarge your border, as He has promised you, and you shall say, 'I will eat flesh' because your soul longs to eat flesh; you may eat flesh, whatever your soul desires. (Deuteronomy 12:20, HBFV).