Unless Target is developing their in-house(or contracted; but with more say over the product than your basic shrinkwrap consumer) against Google Play Services (or, even more foolishly, some OEM's pet extensions); they aren't really all that dependent on Google.
AOSP isn't really designed to be turned into an end-user-ready phone; and lacking Play store, the various Google apps and services, etc. is typically a deal breaker unless you swap in your own (as with Amazon); but if you are treating it as a substitute for the WinCE that historically powered these sorts of fancy-inventory-scanners; or as an easier-to-go-from-BSP-to-graphics-and-a-well-known-platform alternative to rolling a custom Linux frmware; you don't really depend on Google much.
They don't develop AOSP terribly openly, it's pretty much periodic dumps of their project and their plan; but the licensing is open, so you can just keep using it until you can't get hardware for that version anymore, or doing your own security patches gets to be too much; and the AOSP base isn't missing anything particularly glaring for supporting network, barcode scanner, QR/other interpretation from the camera, some dodgy frontend application that supports talking to your inventory system and displaying bits of your web site.
Once you connect something to a network, you aren't going to get nigh-endless support without paying a vendor to care(since the cost of just ignoring security flaws is so much higher); but, especially if you are only using a subset of features, AOSP doesn't exert much control over you if you use it as a base for your firmware.
And, given the specific requirements of retail inventory(durability, battery life, high speed barcode scanning, etc.) you are probably pushing the bounds of what makes sense to try to COTS with just a custom phone case of some sort(unless you are running an operation small enough that the low prices and economies of scale persuade you to forgive a few sins in exchange for being able to buy replacements in quantity one at any cellphone pusher.
At least with POS systems, this seems to have been much what has happened: Square and their imitators blew the bottom out of the market by allowing you to turn normal phone into a cellular card processing terminal(normally a surprisingly pricey item); and there are some mostly small-business focused "iPad embedded in stand/card reader" products; but your Micros and NCR and the like seem to have substantially gone with clearly tablet inspired(and tablet component based, I'm sure they appreciated having touchscreens become cheap and ubiquitous; even if Elo almost certainly didn't) custom hardware running some generic Android based firmware that does nothing except support their application. Once you stamp out enough of them, being stuck with somebody else's product launch cycle, endless changes, and irrelevant features just doesn't make up for the low cost of small quantity orders.