From a lore standpoint, yeah, absolutely. Granted i don't know much about TES lore, but from what I understand only a dragonborn can light the dragonfires and become emperor, so yeah, in theory you could do that. I doubt the game will let you, but the lore allows it.
I'd say the Dragonfires became irrelevant when Martin sacrificed himself and the Amulet of Kings to summon a deus ex machina. Since the Dragonfires could only be lit with the Amulet of Kings, that's why you had to go through so much trouble to get it back when the Mythic Dawn took it from Weynon Priory (Which wouldn't have happened if Jauffre hadn't wanted to keep the Amulet with him, nice job breaking it Jauffre. But I guess he can't be blamed for not wanting to leave such an important item in the hands of a prisoner who might not even survive the trip to Kvatch.) If the Amulet wasn't needed to light the Dragonfires, you could have just gone on to the Imperial City after saving Martin and the whole crisis could end there. Still, that doesn't mean the player can't be the Emperor, it just means that if a new emperor rises, it probably doesn't have to be a Dragonborn.
Still, I highly doubt the player will become the Emperor, while technically, it could work from the lore standpoint, it could cause a problem with the fact that players are free to create their own characters. Letting the player become the Emperor would mean that the player character must be given an identity, even if only in potential sequels, and that's something Bethesda has always tried to avoid. Besides, from a narrative standpoint, it would seem rather awkward as the Empire only just collapsed, okay, in terms of in-game timelines, the events that lead to its collapse actually started too hundred years ago, but in real life, we only just saw the death of the Septim line in the last game. Usually when you do things like that, you don't have the Empire rise again to its former glory in the next game, you do something with it. Either you use it to explore the effects of lacking a unified government on Tamriel, or you have a new power rise in it's place, and I think the latter possibility deserves a story dedicated to it, in Skyrim, the story is about fighting dragons and stopping Alduin from eating the world, I think having the player become Emperor in it would come of as saying "Thank you for stopping the big bad evil guy from destroying the world and bringing peace to Skyrim, and by the way, you're our new Emperor now." Finally, would you want your continent wide Empire to be ruled by someone who is constantly neglecting his duties to go out and loot dungeons and kill rats for the Fighter's Guild?