The verb would be "feels" if the subject of the sentence were "he;" however, the form "him" is never the subject, and in this case is the object of make. That's the short answer.
What feel exactly is in this sentence is complicated, and I'm not sure I understand it. But here's my guess: "feel" is an auxiliary infinitive used with "make." Unlike most infinitives, it is not expressed with the word "to" in front of it. I'm not sure why, but it is an infinitive. Consider this sentence with "cause" instead of "make."
Walking causes him to feel good.
In this case feel is also an infinitive, expressed normally.