There are many people who doubt, if Buddha was really a Avatar of Vishnu. But, in our mind we have no doubt that Buddha was indeed an Avatar. Let me explain why. I am taking little longer route, so have patience.
As we all know, Krishna was all powerful, and could defeat any army, alone, without much effort. If he wanted, he could have finished Mahabharata war in few minutes, by killing whole kaurava army. That way, he could have saved whole bloodshed of ‘good’ people from Pandava army, including his nephew Abhimanyu, and all sons of Draupadi.
Then why do you think, he did not do it? Instead chose to only provide tactical support and advice where needed, and let Pandavas fight their war.
Because, people value things achieved by their effort. Things achieved by sacrificing sweat and blood, winning something having it almost lost, has lasting effects on society, compared to something given free.
It is said that more than thousand year after Mahabharata war, there was peace in the region, without any significant fights or war.
So, main aim of Krishna Avatar, apart from getting rid of bad people like Kansa, Shishupal etc, was to politically unite India (aaryavrata) and stop useless wars among themselves, and bring peace and prosperity to society.
But, there is one more thing Krishna started, and maybe left unfinished, was moving people away from Vedic religion to Vedanta. With Bhagavad Gita, Krishna stressed importance of it, but very few got to know that in his lifetime. Even when Ved Vyas wrote it down it still remained not-so-popular. There were some sections of society following it, but majority People continue to do too many yadnyas, focusing on animal sacrifices, even changing vedangs to prove their point.
So, how do we solve this? It seems, no one cared about ‘free’ knowledge given in Bhagavad Gita.
That’s where Buddha avatar played the role.
Buddhism showed people importance of Vedanta. It ‘forced’ reforms in Hinduism. It was a roundabout way, similar to Krishna not fighting Mahabharata war, and letting people do their own fight. Hinduism was almost marginalized, and was on the way to extinction. People fought their own war and got it back in new improved Vedantic form.
Impact of this is so much that, for more than a millennium now, Vedanta is firmly rooted in Hinduism, and is mainstream, and we can’t even think of going back to Vedic practices.
Vedantic practices like Bhakti, in the form of Bhakti movement, has helped us firmly stay Hindu even after centuries of Muslim and British rule. No other country has done that. It only happened in India, and one of the main indirect contributor to that is Buddha Avatar.
In summary, even though on the face of it, Buddha Avatar does not seem like incarceration of Vishnu, but it did everything that was needed to get back us on track (may be little indirectly) meeting his pledge of “Yada yada hi Dharmasya…”.
So, indeed, Buddha is THE 9th incarnation of Vishnu.