Can I say “the US people”

This usage is awkward and uncommon, but for reasons of idiom rather than grammar. "American people" is far more common.

To get a few technicalities out of the way: "US" is not technically a word, but an initialism. This shouldn't matter, though, since we use such abbreviations as if they're words ("I got a call from some FBI agents"), and they can definitely fill syntactic roles. It's also worth noting that various style guides can make different recommendations about the styling of initialisms, and some might want "U.S." rather than "US"; this also helps avoid confusion with the pronoun us.

You could use similar initialisms to modify similar nouns without idiomatic difficulty: "The NASA scientists had a breakthrough." You could even use "people" in this construction with groups that are not political or demographic, though this usage would be informal: "The CEO was on the phone with the Google people for an hour." This would be equivalent in meaning and tone to "those folks from Google."

There are also many examples of "U.S." used as an adjective to modify many nouns, just not "people": "U.S. Treasury," "U.S. interests," "U.S. citizens."

The difficulty is that many proper nouns for countries, people groups, and ethnicities have associated adjective forms. These often apply to people belonging to that group—"demonyms," words that refer to a person or people from a place. America -> American, China -> Chinese, Midwest -> Midwestern, Slav -> Slavic. When these forms exist, they insist on being used. "The America people," "the Midwest people," etc. are not idiomatic. However, some groups have no such form, and these use the same form for noun and adjective: "The Inuit elders," "the Uyghur population," "the Easter Island residents." (I note that most of these examples are demographic groups or geographical entities below the level of sovereign nations; most countries seem to have adjectival forms. Also, some groups have adjectival forms that are not always used: Iroquoi and Iroquoian can both be found in dictionaries as adjectives meaning "of or relating to the Iroquois people."

... And here's where attempts to explain break down. I was going to conclude by claiming that when an demonym exists it's always preferred. But the fact is, "U.S." is regularly used as an adjective. Some phrasal constructions feature it; "U.S. interests" has supplanted "American interests" since the 1970s. In most cases that a country has an initialism that is widely used in English, that initialism also serves as an adjectival. This Wikipedia table lists U.S. as an adjectival for the United States of America along with American, and UK along with British. When the USSR existed, usages like "a USSR spy plane" were common.

N-grams are a notoriously inexact way to interrogate usage, but if we graph "American people" along with "the U.S. people" (including the to filter out "us people," using the pronoun us) and "people of the U.S.", "American people" by far outstrips the other two. "British interests" dwarfs "UK interests."

Language isn't governed by rules, it's described. It is safe to say that demonyms are generally favored over initialisms as adjectives, except in a few specific phrases and usages like "U.S. interests." And the remarkable lack of examples of "U.S. people" makes it safe to say that this construction simply isn't idiomatic.