analyzing the chord progression in a four part choral piece

In short, it's a song in four parts - for example, Soprano, mezzo, Tenor, Baritone - sung by a choir or vocal group.

And you start with "what is the chord progression?" paying particular attention to cadences and modulations.

A :-)

The four parts are any four voices that move independently of each other.

As for how to analyze the piece. This varies by the skill level of the person looking at the music. The more practice you have, the more things you can just "see."

One method:

1. move all of the voices to the same octave range, clearly label the voice is singing the melody and realize that it can be ignored or included as needed to produce the simplest explanatory framework.
2. re-arrange the tones in each resulting chord to be a root inversion in thirds.
3. label the resulting chords I ii iii IV V etc. don't use power chords or suspensions in this step.
4. recognize that few songs have a new chord every beat, so look for opportunities to assume missing tones to simplify the chord progression.
5. look for opportunities to assume one or more of the tones functions as an alteration
6. look for opportunities to assume one or more of the chords is a substitution for a neighboring chord (perhaps a substitute dominant in place of a dominant chord, for example).

Remember that all of this happens in multiple contexts: the local context of the immediate phrase; the large context of the immediate section; and the context of the full work.