Here's footage of MAME's emulation for analysis.
I'll have to watch it later when I get some time (IRL has kept me very busy, so I haven't had time to do much gaming-wise lately), but my short-term answer is: as long as the game plays at the same speed as the original hardware, and the enemies, drops, and gameplay is identical, than it should be allowed.
Is this the case? That's the analysis we'd have to do, but you could check it yourself: put the game up alongside a run on original hardware. A good place to do this would be after you slay the final enemy, you could put this MAME emulator against the real hardware. The cutscenes should play frame-by-frame identically, taking into account any variables such as score, or where Link is standing when he kills the final enemy.
There was another emulator that has a tomahawk in the first room of the game (on Pica Pic IIRC). the real hardware cannot replicate this. That's an example of a reason it WOULD NOT be allowed. But there's always the possibility of another category if that's the case.
TL;DR until one of the mods (or anyone really) can do the analysis and make a conclusion, if this is the only way you can play the game, I still encourage you to do so! If it's determined that this is an unfair advantage to play over the real hardware or another emulator, we can always just make a seperate category. Don't let this stop you from speedrunning, if you truly want to!