No, that's not an orchestra
Even though it sounds as if there is a live orchestra playing on the track, no instruments were used at all.
Co-writer Walter Afanasieff revealed that the music was all arranged on his computer and the only real elements of the song were Mariah's voice and those of her backing vocalists.
He told Variety: "We did other songs during the sessions with the band. But there was never a version [of "All I Want for Christmas Is You"] with a band.
"I think we might've tried it, and it just didn't sound as good. It needed to be brighter, more fun and more of what I thought a Christmas song should be on the radio, so I kept all of my synth/sequencer parts, computer parts."