In a way, what 'the chord' is at a given time is, by definition, the notes that are being played at that time - so yes, the accompanying instruments will be playing notes in the chord - but only because the chord is the notes that are being played... (etc. etc., round in circles...)
If we followed that idea strictly though, it would mean that 'the chord' could potentially be changing every time a note stopped or a new note was played. What tends to happen is that people consider the notes in the chord to be those notes that are being played consistently or repeatedly (so not every note played is considered to be part of the chord) for a period of time until it becomes clear that the set of notes being played (and especially the lowest note) has changed, signifying a new chord.
If we flip the logic one more time... yes, it is fair to say that accompanying instruments will typically be playing the notes in the chord, although there might be other ornamental or decorative notes in there.
If we extend this idea - it would even be possible to consider that even an unaccompanied solo instrument was following a chord progression, for example if it was arpeggiating notes in from a sequence of chords in turn.
If you read the chord progression from a transcription, and then listen to a recording, you will sometimes find that not all of the notes in the written chord are present or prominent in the recording. This can because there's a presumption towards writing triads - e.g. C Major - but some modern songs survive on sparser harmonies, especially bare fifths (such as power chords) or sometimes a bassline may play a harmonic root that isn't expanded into any particular chord.