I am not sure any printing service goes as low as 16 pages, not in hardcover, anyway.
While KDP/Amazon is not strictly (or only) a printing service, they have a minimum of 24 pages for softcover editions and 75 for hardcover.
Lulu.com is a printing service (that does also handle some direct sales), and is vague about page minimums, having several formatting recommendations that include "0-n pages." I can't find an absolute, pro-level list of layout points, so I would not be surprised if they have an effective minimum, especially for hardcover, that would result in blank pages at the end. (Such as 8 blank pages, an effective minimum of 24 pages for binding.) If you're looking for professional publishing quality for purchase of author (resale) copies, I will just note that Lulu has been my only non-KDP choice for quite some time, and is likely to be the most cost-effective all around.
It's possible some of the retail/amateur providers like Canva or CafePress might have suitable options, but they are likely to be quite expensive in unit cost.
All that said, a 16-page book is pretty much off the bottom of reasonable print sizes. Even most children's books have 32 or so. This is much more true of hardcover; very slim softcover books can use saddle-stitching (stapling) at a cost-effective point. But to go to perfect bind or hardcover will probably see much higher costs because of floor expenses for that kind of production.
Unless you have an unlimited budget, as well, it's best to choose one of their standard — and not extra-cost — trim and binding sizes, and format your book to fit that. Some providers will print odd-sized books, but usually with very high special-job fees.
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