This one isn’t the Buddha.
It reminds me of something from Buddhaghosa’s Visuddhimagga, or “Path of Purification,” which is from 1,000 years after the Buddha:
There is no doer of a deed
Or one who reaps the deed’s result;
Phenomena alone flow on—
No other view than this is right.
I’m pretty sure that our quote is a paraphrase or alternative translation of that passage.
Incidentally, the words are not Buddhaghosa’s. He makes it clear that he is quoting some unnamed “ancients.”
Another passage with similar resonances is also something Buddhaghosa is quoting anonymously:
For there is suffering, but none who suffers;
Doing exists although there is no doer.
Extinction is but no extinguished person;
Although there is a path, there is no goer.
There are probably similar teachings in the Mahayana Sutras. But there’s nothing quite like this in the early scriptures.
There is in the Phagguna Sutta an interesting series of passages like this:
“Who, O Lord, clings?”
“The question is not correct,” said the Exalted One, “I do not say that ‘he clings.’ Had I said so, then the question ‘Who clings?’ would be appropriate. But since I did not speak thus, the correct way to ask the question will be ‘What is the condition of clinging?’ And to that the correct reply is: ‘Craving is the condition of clinging; and clinging is the condition of the process of becoming.’ Such is the origin of this entire mass of suffering.”
So here the Buddha declines to talk in terms of “one who clings” and instead talks about how clinging comes to be, and what it leads to. This isn’t a million miles away from saying that “phenomena alone roll one” without a “doer.” But that still doesn’t make the quote in question a genuine quote from the Buddha.
He also comes close to the philosophy of those “Ancients” in the Kalaka Sutta, where again there are a number of passages similar to this:
[T]he Tathagata [i.e. the Buddha], when seeing what is to be seen, doesn’t construe an [object as] seen. He doesn’t construe an unseen. He doesn’t construe an [object] to-be-seen. He doesn’t construe a seer.
This seems to be that is called a nondual perspective, where he doesn’t conceive in terms such as “one who sees,” or “a thing that is seen.” And yet, seeing happens.
So, this quote is Buddhist, but we can’t say it’s from the Buddha.