Well, you're lucky, because there are a lot of people making very high quality NES cartridge already (the NES homebrew scene is very vibrant compared to the SNES scene).
I can recommend you to shopping at Infinite NES Lives, they are selling blank cartridge (loads of mappers variants available) and a tool to "write" the game on the cartridge:
http://www.infiniteneslives.com/nessupplies.php
(we actually used their flasher to write the SNES cartridges of Yo-Yo Shuriken, as it works for SNES too!)
With that you can make your own physical release too! :)
That feels like a bit of cheating if you are not designing & soldering your own cartridge :P
I wonder how this was like 40 years ago. Would developers use services similar to this: find a company that makes the mapper you use, send them your binaries and leave them to production details? Or would you design your own boards and use a service to create and assemble them? Perhaps very large companies could even afford build their own boards.
There are some complex/unique mappers out there which makes me believe at least some companies out there were designing those mappers and building their own boards
Well, for my project I did partner with someone to make the PCB, as I have about 0 electronics skills (maybe someday...). People like INL or Catskull are usually one-man operations, and they design their own PCB from scratch, so we are still in the "homebrew" spirit ;). But I get your feeling of wanting to build your own PCB on your own - I think you may find some schematics / documentation online to get you started if you want.
During the 80's-90's, AFAIK, Nintendo was the only company authorized to manufacture cartridges. So any publisher / developer would have to buy hardware from them. And you wouldn't be able to make a small lot, but you had to buy tens thousands or maybe hundreds thousands of units - this was a large scale business!
From what I've read, the process was indeed: build the final rom, have it pass a thorough QA session at Nintendo, pay for the cartridges and Nintendo will build them for you.
For the NES era, several companies did manage to make their own mappers. For example Konami designed a mapper to add extra audio channels in a NES cart. But, for some reasons, Nintendo didn't allow third party mappers in US and European releases. That's why only Japanese version of games like Castlevania III have such mappers "not made by Nintendo".
And of course, many unlicensed companies did create their own cartridges from the ground up, like Codemaster and Color Dreams/Wisdom Tree. They were not big companies at the time. So I think even middle sized ones could create their own cartridges and mappers if they had the skill inhouse!
I can recommend you to shopping at Infinite NES Lives, they are selling blank cartridge (loads of mappers variants available) and a tool to "write" the game on the cartridge: http://www.infiniteneslives.com/nessupplies.php
(we actually used their flasher to write the SNES cartridges of Yo-Yo Shuriken, as it works for SNES too!)
With that you can make your own physical release too! :)