Obviously there could be factors at play here that you have not included, but based on the facts in your question, the simple answer is:
The client is right, and your team are being difficult for some reason
Ascertaining the reason your team are demanding the .ai files. Which they certainly do not need to print, is the key here to avoid conflict.
Start by understanding the workflow:
The client is using Adobe Illustrator to create the design. That is entirely normal. But he could be using Microsoft Paint or Gimp, he could use Word (unusually!) but the fact is he could use whatever tools he finds fit for purpose.
The people he sends his design to, your team, or other people, may well not have those tools, so for many years the standard workflow has been to save the finished work as a PDF file and send that, since anyone can use a PDF file. It's an "industry standard".
The fact that your team are insisting on receiving the .ai file is NOT standard practise, but of course they are doing it for one reason or another. Here are some guesses, least likely to most likely.
Possible reasons your team want the .ai file:
- (Least likely) Your team simply like upsetting people with pedantic and in-efficient behaviour. Maybe they don't like the client, or you, and they want to make life as difficult as possible.
- They are used to printing from .ai files, or are ill-educated about printing from PDF files (very unusal if they are professionals)
- (Most likely) They want to change things AFTER they receive the file... maybe colours, maybe text...maybe anything - and this is probably at the heart of what is annoying the client.
Why the client is annoyed.
- It is not standard to have to turn over working files (.ai files)
- He doesn't want people changing his design - possibly ruining it, introducing typos he'll get blamed for etc etc after he has finished
- He doesn't want people adapting his work to new designs for which he will not be paid.
- It's just plain annoying and he thinks your team are trying to upset him on purpose.
- His working files might be "untidy" (immaterial if the finished output is fine) so he'll have to spend much extra time making them fit for a third-person to inspect, and possibly criticise.
The way forward
Tell the client you understand and sympathise with his situation and you aim to resolve it.
Ask your team WHY they want the .ai file and deal with the answer accordingly - perhaps there really is a legitimate reason. If so you would htn have to talk with the client and explain that reason to him.