Hi OP, due to the recent time period in question, it may be difficult to get an answer in this subreddit which will comply with the rules, especially the rule against personal anecdotes. While your question can certainly be asked here, and could get a compliant answer, It may be worth x-posting to a sub with looser sourcing rules, for example one specializing in computing or games development.

would try r/nintendo and maybe ask some of the dev studios the had been involved during that period

or /r/nintendoofamerica in a few weeks, we might have some proper sources by then

For most of its life the NES didn't have an official developer kit (though there were third party kits focused around testing), however this wasn't a major issue as the NES didn't not have an operating system like a PC. It simply ran the programme on the cartridge with the language of the programme being assembly code (basically tell the console when and how to load things into memory). Most games were written on home computers of the time like the C64 which had similar specifications as the NES.

At this point we have to move onto getting your game onto the NES. The NES was a closed system so you needed be permission from Nintendo and sign up to a licensing agreement, usually via a publisher. Your game would have a meet a high standard of quality and could be censored to removed certain themes. Once that was all done then Nintendo would supply the cartridges to you for programming at around $35 a cartridge.

However your game may be delayed because in 1987/88 the computer industry is facing a shortage of computer chips due to high demand resulting in Nintendo cutting back on cartridge deliveries.

In all developing for the NES (officially at least) was outside the ability of most solo programmers who focused on the multiple home computer formats of the time.