Phonons are the minimum quantum excitation of vibration within a material. Just as harmonic oscillators have quantum energy levels, so do materials. You are correct -- they are an emergent phenomenon.
In the case of phonons, they can carry momentum, they travel like particles. A sibling comment is correct -- they are technically quasi-particles [1]. It is okay to think about a phonon field within a material. That field is not fundamental, in the sense that an electron is presently-believed to be fundamental, but that doesn't keep it from being a very useful way to think about condensed-matter physics.
In the case of phonons, they can carry momentum, they travel like particles. A sibling comment is correct -- they are technically quasi-particles [1]. It is okay to think about a phonon field within a material. That field is not fundamental, in the sense that an electron is presently-believed to be fundamental, but that doesn't keep it from being a very useful way to think about condensed-matter physics.
[1] https://en.wikipedia.org/wiki/Quasiparticle