Search is (or should be) a part of your Information Architecture. IA doesn't exist in a vacuum, but it's bound but the context and the available objects and modules (pages, sections, features and so on). IA has hierarchies and taxonomies that have to be clearly defined in order to provide a pleasant experience.
This being said, you can see that, a search on a header has a really high hierarchy, it's a visual reminder that yells "hey, I'm on top of everything!". This means users will be compelled to use it.
A search on a navigation bar will be part of the site structure. Users may not understand they can look for products, but that the search bar is to look for pages or sections, like in any other site. Furthermore: a search box inside a nav on mobile will probably disappear and users will need to interact with another element in order to find it. Assuming they know it's there!
Online shopping is search centric, and that's a given. But of course things are not as easy as I mention them. I'll give you an example: take a look to some really big online shopping sites:
Do you see the pattern? Not only all of them have search as the predominant element, but navigation bars are really scarce, or limited to buying process, or simply disappearing. Think on Google, AirBnB, Trivago, Bing and other sites which are basically a search box (AirBnB added more elements now, but still has the search box as the focus). And the reason is easy: users want to search, then... why distract them with stuff that has no relation with what they want to do? If in doubt which approach works better... ask Yahoo.
If search is this important for your stakeholders, then it has to be in the header. Millions have been spent in research for this pattern, so they can benchmark other successful sites and learn, no need to reinvent the wheel