In the vast majority of the Vedic hymns (Samhitas), there is nothing to critique, as they consist of exaltations of gods, prayers, and rituals, which do not have philosophical, scientific, or historical content that can be examined or tested.
A rare exception to this pattern is the famous 129th hymn from book 10 of the Rigveda, which usually bears the heading "Hymn of Creation":
Rigveda - Book 10, Hymn 129
1 Then was not non-existent nor existent: there was no realm of air, no sky beyond it. What covered in, and where? and what gave shelter? Was water there, unfathomed depth of water?
2 Death was not then, nor was there aught immortal: no sign was there, the day's and night's divider. That One Thing, breathless, breathed by its own nature: apart from it was nothing whatsoever.
3 Darkness there was: at first concealed in darkness this All was indiscriminated chaos. All that existed then was void and formless: by the great power of Warmth was born that Unit.
4 Thereafter rose Desire in the beginning, Desire, the primal seed and germ of Spirit. Sages who searched with their heart's thought discovered the existent's kinship in the non-existent.
5 Transversely was their severing line extended: what was above it then, and what below it? There were begetters, there were mighty forces, free action here and energy up yonder
6 Who verily knows and who can here declare it, whence it was born and whence comes this creation? The Gods are later than this world's production. Who knows then whence it first came into being?
7 He, the first origin of this creation, whether he formed it all or did not form it, Whose eye controls this world in highest heaven, he verily knows it, or perhaps he knows not.
Analyzing this hymn, beginning with verses 1 and 2:
Rigveda - Book 10, Hymn 129
1 Then was not non-existent nor existent: there was no realm of air, no sky beyond it. What covered in, and where? and what gave shelter? Was water there, unfathomed depth of water?
2 Death was not then, nor was there aught immortal: no sign was there, the day's and night's divider. That One Thing, breathless, breathed by its own nature: apart from it was nothing whatsoever.
Verse 1 begins with "Then was not non-existent nor existent". In logic, there are three fundamental axioms which are required to begin to use reasoning. One of these is the law of the excluded middle, which states that for any proposition P, either P is true, or not P is true. Verse 1's statement violates the law of the excluded middle for any consistent use of the word "existent", as something either exists, or it does not exist. Therefore, there could never be a time when there "was not non-existent nor existent".
Verse 2 says "Death was not then, nor was there anything immortal". However, verse 2 also says, that the "One Thing", "breathed by its own nature", was present. This "One Thing", then, "existed", and was "immortal", using any standard definition of those words, which contradicts verse 1, and the preceding part of verse 2.
A Hindu may respond with something to the effect of - This describes a situation that transcends existence, and transcends mortality and immortality. No, it does not. Nothing does. The concept of existence is impossible to transcend, without resorting to nonsense. Likewise, "immortality" is a concept which encompasses something that is not subject to decay - which certainly would describe a transcendant, self-subsisting being like God.
Verses 3 through 5 describe the creation of the universe.
Verses 6 and 7 then speculate on the origin of the creation:
Rigveda - Book 10, Hymn 129
6 Who verily knows and who can here declare it, whence it was born and whence comes this creation? The Gods are later than this world's production. Who knows then whence it first came into being?
7 He, the first origin of this creation, whether he formed it all or did not form it, Whose eye controls this world in highest heaven, he verily knows it, or perhaps he knows not.
Verse 6's statement that "The Gods are later than this world's production" is not surprising to a Hindu, nor does it refute Hinduism, as some who are new to studying Hinduism may suppose. Most Hindus believe in either a Supreme God - such as Vishnu, or Shiva - who is eternal, and created the other gods, or, they believe in a Non-Dualistic view of the universe in which everything is a manifestation of Brahman.
However, verse 6's question, "whence comes this creation?", is highly significant, as verse 7 answers by stating that either the "One Thing" mentioned in verse 2, who created the universe, knows the answer, "or perhaps he knows not". The suggestion that perhaps even God does not know something would be considered blasphemous in most religions, as it would be suggesting an imperfect, non-omniscient God, which would be slander to God if God were omniscient - something seen as foundational in almost every other religion.
Some translations render the last line "If he does not know, then who knows?", rather than "or perhaps he knows not", as above. This would resolve the apparent logical conundrum presented by the final two verses, if it is a legitimate translation.
The Sanskrit for the portion in question is:
Looking at the above, it would seem that "perhaps he knows not" - would be the more straightforward translation, rather than "If he does not know, then who knows?" - which reads into the text quite a bit.
Translators who render it as something similar to "perhaps he knows not" include:
Therefore, there is good evidence for the reading suggesting a non-omniscient concept of God. However, the reading is contested.
Regardless of one's view of verses 6 and 7, the critique of verses 1 and 2 stands. When the monotony of the Vedas is broken by an extremely rare philosophical tangent, the result is contradiction giving way to nonsense. And, very possibly, the suggestion of a non-omniscient God.
The Samhita portions of the Vedas, with their endless hymns to gods that most Hindus do not even worship anymore (Indra, Agni, Mitra, Varuna), their descriptions of rituals no one does anymore, their endless exaltation of an all-important drink which is now lost (Soma, mentioned thousands of times), and the denial of their access to the lower Castes (see Appendix I), cannot really be said to be the foundation of modern Hinduism, which is mostly a reflection of the Bhakti movement and the Puranas. The ritualism of the Vedas, or even the Non-Dualistic philosophy of the Upanishads, is eschewed by most Hindus in favor of a devotional Hinduism.
Reading the Vedas, it is easy to understand how such a development was ripe for taking place. The Vedas are mostly boring, mostly irrelevant, and even in their rare venture into philosophy, they falter.
It is worth noting that, historically, the Vedas were not even allowed to be read, touched, or listened to by members of the lowest Caste (Shudra), or those outside the Caste System (Dalit, untouchable):
Gautama Dharmasutra - Chapter 12
4 Now if (A Śūdra) listens intentionally to (a recitation of) the Veda, his ears shall be filled with (molten) tin or lac.
5 If he recites (Vedic texts), his tongue shall be cut out.
6 If he remembers them, his body shall be split in twain.
Brahma Sutras - Book 1, Chapter 3
38 And on account of the prohibition in Smriti of (the Sudras) hearing, studying and understanding (the Veda) and performing Vedic rites (they are not entitled to the knowledge of Brahman).
Manusmriti - Chapter 4
98 After this, (the Brāhmaṇa) shall diligently read the Vedas during the bright fortnights, and all the subsidiary sciences during the dark fortnights.
99 He shall recite, not indistinctly, nor in the proximity of Śūdras; nor shall he go to sleep again, at the end of night, when he is tired after having recited the Veda.
Vishnu Smriti - Chapter 30
3 During the period (subsequent upon the ceremony of Upakarman and) intermediate between it and the ceremony of Utsarga, the student must read the Vedangas
10 (He must not study) in a village in which a corpse lies
11 Nor during a battle
12 Nor while dogs are barking, jackals yelling, or donkeys braying
13 Nor while the sound of a musical instrument is being heard
14 Nor while Sudras or outcasts are near