I have often heard from many people that Kshatriyas are allowed to partake non-vegetarian food. Some people who believe Sri Rama took meat (this is due to certain controversial verses present in the Valmiki Ramayana) defend Him by using the argument that Kshatriyas are allowed to consume meat. Do any scriptures allow Kshatriyas to consume non-vegetarian food?
The Smritis do not allow any of the twice borns to eat meat that is not first offered to Gods or Manes in sacrifice.
Manu Smriti 5.31. ’The consumption of meat (is befitting) for sacrifices,’that is declared to be a rule made by the gods; but to persist (in using it) on other (occasions) is said to be a proceeding worthy of Rakshasas.
Manu Smriti 5.32. He who eats meat, when he honours the gods and manes, commits no sin, whether he has bought it, or himself has killed (the animal), or has received it as a present from others
Manu Smriti 5.33. A twice-born man who knows the law, must not eat meat except in conformity with the law; for if he has eaten it unlawfully, he will, unable to save himself, be eaten after death by his (victims).
So, unlawful consumption of meat is condemned like that in every Hindu scriptures, including the Tantras.
When life is in danger, then of course, one is allowed to eat non-sacrificial meat
Manu Smriti 10.106. Vamadeva, who well knew right and wrong, did not sully himself when, tormented (by hunger), he desired to eat the flesh of a dog in order to save his life.
The verse that is given in this answer is also found in the Manu Smriti:
5.17. Let him not eat solitary or unknown beasts and birds, though they may fall under (the categories of) eatable (creatures), nor any five-toed (animals).
5.18. The porcupine, the hedgehog, the iguana, the rhinoceros, the tortoise, and the hare they declare to be eatable; likewise those (domestic animals) that have teeth in one jaw only, excepting camels.
But, that one is not recommended to eat them without any rituals is already made clear from the other Manu Smriti verses given above.
In Valmiki Ramayan: Kishkindha Kanda: Sarga 17, Vali (Bali) says:
पञ्च पञ्च नखा भक्ष्या ब्रह्मक्षत्रेण राघव।
शल्यकः श्श्वाविधो गोधा शशः कूर्मश्च पञ्चमः।।4.17.38/39।।
O Rama, Brahmans and Kshatriyas are permitted to eat only the 5 kinds of five-nailed animals, viz., the porcupine, the hedgehog, the alligator, the rabbit and the tortoise. [English translation is taken from source 1 & source2]
Scriptures recommend to follow some rituals before eating food which is already discussed in this Question:
According to Rāma, if you are a king (kṣatriya) you are allowed to hunt deer for both its flesh and also as a sport.
मांस हेतोः अपि मृगान् विहारार्थम् च धन्विनः |
घ्नन्ति लक्ष्मण राजानो मृगयायाम् महावने || ३-४३-३१
31. lakSmaNa = oh, Lakshmana; raajaanaH mahaa vane mR^igayaayaam = kings, in great, forests, in games of hunting; maamsa hetoH api = flesh, for the sake of, even; dhanvinaH [dhanvanaH] = archers [of bows, or, to use bows in sport, archery]; vihaara artham ca = sporting, for purpose of, also; mR^igaan ghnanti = deer, they kill.
Kings pursuing games of hunting in great forests, oh, Lakshmana, will be felling deer either for the sake of flesh, or just for the purpose of sporting archery. [3-43-31]
Bibek Debroy translates it little differently:
Can you name anyone whose mind will not be tempted by this deer? His form has the complexion of molten gold. There are many kinds of celestial jewels. On seeing him, whose mind will not be filled with wonder? On hunts, kings roam around in the great forest. O Lakshmana! Wielding bows, they kill deer for the sake of their flesh. They exert themselves in the great forest and collect many kinds of minerals, gems, jewels and gold. All those riches make men prosper. O Lakshmana! Everything thought of in the mind enhances Shukra's treasure house. O Lakshmana! If a person desires artha and goes around unhesitatingly collecting that artha, those who know the sacred texts about artha say that this artha is true artha. With me, the slender-waisted Vaidehi will be seated on half of this deer's golden and gem-encrusted hide.
Chapter 3 (Sarga 41)
Bhishma in the 116th chapter of Anushasana Parva of the Mahabharata, declares that Kshatriyas don't commit any sin by eating meat:
"Bhishma said, 'It is even so, O mighty-armed one, as thou sayest. There is nothing on earth that is superior to flesh in point of taste. There is nothing that is more beneficial then flesh to persons that are lean, or weak, or afflicted with disease, or addicted to sexual congress or exhausted with travel. Flesh speedily increases strength. It produces great development. There is no food, O scorcher of foes, that is superior to flesh. But, O delighter of the Kurus, the merits are great that attach to men that abstain from it. Listen to me as I discourse to thee on it. That man who wished to increase his own flesh by the flesh of another living creature is such that there is none meaner and more cruel than he. In this world there is nothing that is dearer to a creature than his life. Hence (instead of taking that valuable possession), one should show compassion to the lives of others as one does to one's own life. Without doubt, O son, flesh has its origin in the vital seed. There is great demerit attaching to its eating, as, indeed, there is merit in abstaining from it. One does not, however, incur any fault by eating flesh sanctified according to the ordinances of the Vedas. The audition is heard that animals were created for sacrifice. They who eat flesh in any other way are said to follow the Rakshasa practice. Listen to me as I tell thee what the ordinance is that has been laid down for the Kshatriyas. They do not incur any fault by eating flesh that has been acquired by expenditure of prowess. All deer of the wilderness were dedicated to the deities and the Pitris in days of old, O king, by Agastya. Hence, the hunting of deer is not censured. There can be no hunting without risk of one's own life. There is equality of risk between the slayer and the slain. Either the animal is killed or it kills the hunter. Hence, O Bharata, even royal sages betake themselves to the practice of hunting. By such conduct they do not become stained with sin. Indeed, the practice is not regarded as sinful.
As per Lord Krishna the highest authority, any killing, especially animals, done by kshatriya is not exempt from sin.
We find multiple examples in Srimad Bhagavatam, e.g. King Prachinabarhi who killed animals during sacrifice and Narada showed him a dark future.
All these animals are awaiting your death so that they can avenge the injuries you have inflicted upon them. After you die, they will angrily pierce your body with iron horns.
Srila Prabhupada gives a masterly commentary as always on Narada's words.
in this verse in Srimad Bhagavatam, Lord Krishna instructs Maharaja Muchkunda as follows:
Because you followed the principles of a kṣatriya, you killed living beings while hunting and performing other duties. You must vanquish the sins thus incurred by carefully executing penances while remaining surrendered to Me.