Piet, this sounds like an odd problem that I've not heard about before. It's got me stumped but I don't send much to be printed externally nowadays.
It might help someone else to diagnose what's wrong if you can supply the following details:
* What PDF version are you exporting to?
* When exporting are you embedding PDF/EPS files? (This is an experimental feature.)
* What output type did you use when exporting? (E.g. Screen/Web, Printer, Greyscale?)
* What compression options did you use?
* What type(s) of illustrations are missing? (E.g. Only SVGs? Only PNGs? etc.)
* Are the missing illustrations on the same layer as other stuff which is correctly printed?
* What software is being used to print at the copy shop? (And does it have any limitations?)
And two other things which probably aren't relevant but might be worth looking at are:
* Are the missing illustrations single-colour or multi-coloured?
* Have you got colour management on or off? (If on, what settings have you used?)
I only suggest these last two as you may be using a colour which the copy shop printer can't print - but in these modern times the probability of that is very small, if not zero, in most cases. (I sort of remember talking to someone about this 20+ years ago, but I can't be sure. I thought I'd throw it into the pot in case it rang a bell with someone.)
I've no idea if any of this will help someone else to help you but it's best to provide all the info you can, especially when the problem is strange.
I hope you find a solution.