I did some research with materials in my collection, and I think I can provide a thorough answer (probably more thorough than necessary, but maybe somebody will find this interesting).
From my research it appears that the orchestra for the original Broadway production was scored for 20 pieces. I would think that the scores currently available for rental from MTI would maintain the same number of personnel. It's possible that with the changes in the show in the past 10 years (the extra dance in "NYC" the revised placement of "I Don't Need..." etc.) that the orchestrations might have been changed to support more synthesizer players and therefore less personnel.
Violins 1, 2, 3, and 4
Bass (should be an acoustic upright bass, but now more productions are using an electric bass)
Trumpets 1 and 2
Trombones 1 and 2
Reeds 1, 2, 3, 4, and 5
---As mentioned in another post, the woodwind players have to switch instruments throughout the show. I tried to figure out the breakdown more specifically. I had to do a little guessing to try to assign the reed parts, so this part may not be 100% accurate.---
Reed 1: flute, piccolo, and possibly clarinet
Reed 2: Soprano saxophone, alto saxophone, flute, possibly clarinet
Reed 3: Tenor sax, clarinet, bass clarinet
Reed 4: Alto sax, tenor sax, clarinet
Reed 5: Alto sax, baritone sax, clarinet
Drums (trap set)
Percussion: timpani, bells, xylophone, plus sleigh bells, ratchet, slapstick, triangle, etc.