Question
Why does part of Vaishnavism consider Gautama Buddha to be an Avatara of Lord Vishnu, even though he rejected Vedic wisdom?
Answer
All serious Vaishnavas consider Buddha an avatar of Vishnu, but the question is which Buddha? First, let us prove that the Buddha was indeed an avatar.
Some claim that the Buddha was either an interpolation in the puranas or that the puranas themselves came after the time of the Buddha. Allegedly, this would bring Buddhists into the fold of Hinduism. However, that would make the following verse of Srimad Bhagavatam historically inaccurate.
When the atheists, after being well versed in the Vedic scientific knowledge, annihilate inhabitants of different planets, flying unseen in the sky on well-built rockets prepared by the great scientist Maya, the Lord will bewilder their minds by dressing Himself attractively as Buddha and will preach on subreligious principles. [SB 2.7.37]
Obviously, there were no rockets toward the beginning of the Kali Yuga (traditionally accepted as 3102 BCE but may actually be debatable as seen here), so this verse would not further the puranic "agenda" to subsume Buddhism under Hinduism. Actually, this verse is referring to the Buddha in another kali-yuga as Srila Prabhupada's commentary says.
According to Śrīla Jīva Gosvāmī, the Buddha incarnation mentioned in this verse appeared in a different Kali age. In the duration of life of one Manu there are more than seventy-two Kali-yugas, and in one of them the particular type of Buddha mentioned here would appear.
Part of the actual confusion regarding the Buddha avatar stems from the fact that there are different Buddhas mentioned in the puranas (the above verse being an example). One chapter of the Vishnu Purana also discusses Buddha being sent by Vishnu to delude the atheists, but that Buddha seemed to have emanated the Jain doctrine as well. (I do not currently have the relevant verses.) With that out of the way, let us turn to the Buddha of the present Kali Yuga.
Then, in the beginning of Kali-yuga, the Lord will appear as Lord Buddha, the son of Añjanā, in the province of Gayā, just for the purpose of deluding those who are envious of the faithful theist. [SB 1.3.24]
Based on my research, I have come across three different views regarding the identity of the avatara known as Buddha.
1. The Buddha Mentioned Throughout the Puranas Is Gautama Buddha.
According to Prabhupada's commentary on verse 1.3.24, Gautama Buddha incarnated to stop animal sacrifice since it was being used as an excuse to eat meat (which is typical in this age).
Because the asuras or the so-called scholars of Vedic literatures put forward the evidence of animal-killing in the Vedas, Lord Buddha superficially denied the authority of the Vedas. This rejection of the Vedas by Lord Buddha was adopted in order to save people from the vice of animal-killing as well as to save the poor animals from the slaughtering process of their big brothers who clamor for universal brotherhood, peace, justice and equity. There is no justice when there is animal-killing. Lord Buddha wanted to stop it completely, and therefore his cult of ahiṁsā was propagated not only in India but also outside the country.
As for why the Buddha would go against the Vedas, Prabhupada says that it was due to the circumstances of that time.
Technically Lord Buddha’s philosophy is called atheistic because there is no acceptance of the Supreme Lord and because that system of philosophy denied the authority of the Vedas. But that is an act of camouflage by the Lord. Lord Buddha is the incarnation of Godhead. As such, he is the original propounder of Vedic knowledge. He therefore cannot reject Vedic philosophy. But he rejected it outwardly because the sura-dviṣa, or the demons who are always envious of the devotees of Godhead, try to support cow-killing or animal-killing from the pages of the Vedas, and this is now being done by the modernized sannyāsīs.
2. Gautama Buddha Is Not the Avatar of Vishnu
This article by Stephen Knapp argues that the avatar of Vishnu known as Buddha incarnated before Gautama Buddha. The former is known as the son of Añjanā while the latter was the son of Shuddhodana and Mayadevi. (Someone could claim that Añjanā was an epithet of Mayadevi, the mother, but I do not think she was ever called Añjanā in her life, nor did she have a name change.) The former was born in Kikata, also known as Gaya, while the latter was born in Kapilavastu, Nepal.
Knapp discusses several verses from Srimad Bhagavatam, Buddhist scriptures, and the Amarakosha to prove his point. I will just quote what he said about the Amarakosha.
In these verses, starting with sarvajnah and finishing with munih are eighteen names addressing the original Vishnu incarnation Lord Buddha. The next seven names beginning with Shakya-munistu to Mayadevi-Sutascha refer to Shakya Simha Buddha. The Buddha referred to in the first eighteen names and the Buddha referred to in the later seven names are clearly not the same person. [This clearly indicates that knowledge of the two Buddhas was well known long ago.] In the commentary on Amarakosha by the learned Sri Raghunatha Cakravarti, he also divided the verses into two sections. To the eighteen names of Vishnu Avatara Buddha he writes the words "astadash buddha", which clearly refers only to the Vishnu avatara. Next, on his commentary for the seven aliases of Shakya Simha he writes: "ete sapta shakya bangshabatirneh buddha muni bishete", meaning "the next seven names starting from Shakya-munistu are aliases of Buddha-muni [the human] who was born into the Shakya dynasty."
3. Gautama Buddha Was an Avesha Avatar of Vishnu
The first view held that Vishnu incarnated as Gautama Buddha, and the second view held that Vishnu Buddha was different from the atheistic Buddha. However, in many of his videos on DharmaNation, Sri Dharma Pravartaka Acharya has said that Gautama Buddha was an avesha avatar, which means he was not Vishnu Himself but a being empowered to fulfill his mission. In this video in particular, something that I found quite interesting was that Gautama Buddha never actually taught shunyavada (the doctrine of voidness). He also never claimed to be founding a new religion called Buddhism, and he was in fact following the Vedic scriptures. However, he only taught the negative half of the philosophy: the Buddha said that we are not the body, but he did not teach that we are the soul. There is no evidence in Buddhist scriptures that Gautama Buddha spread shunyavada; this misconception came about when Buddhism became systematized by Nagarjuna hundreds of years later. The Buddha's mission was to transition society away from animal sacrifice toward self-realization and meditation.
Vishnu Purana's aforementioned account of the Buddha can also be considered proof that he was an avesha avatar.
At the end of the day, there are several different viewpoints about the Buddha in Hindu scriptures; after all, this is quite a tricky subject. Not everyone believes in the same thing. I answered a lot more than just what you were asking, but the answer to your main question can be found under the first heading.
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