The best solution that I can think of would be to use a text to speech program such as “Dragon Naturally Speaking” (commercial non-free software) to convert the recording to text, then open the text in a word processor to get a word and character count. From the word count and character count you should be able to make a reasonable approximation of the number of syllables.
Text to speech programs require “training” to get the most accurate results, which may not be possible if the recordings are from different people, but if you use one of the better speech to text programs the default settings will probably be good enough for an approximation.
“Paul L” has done some work in Nyquist for recognising phonemes automatically (https://forum.audacityteam.org/t/preliminary-phoneme-finder-toy/29243/1) but his work is probably not yet at a stage where it is really useful for giving you an answer.