Well, true, Howland Reed wasn’t there when Ned spoke with Lyanna, as far as we know.
However, assuming baby Jon was at the Tower of Joy, Howland obviously would know that Ned traveled there without a baby and left the tower with one. Maybe Ned could have given him the same cover story he gave others (Wylla?), but when Ned and the others kill the three Kingsguard, Ned enters the tower to talk with his sister, and then later Howland and whoever else (Ned at some point says “they” found him with his sister’s body) enter the tower and find Ned, Lyanna’s corpse, and a baby, I think that story would be a pretty hard sell.
The only other possibilities I can think of would be Wylla and Barristan Selmy. Obviously Wylla knows Jon really isn’t her kid (assuming, of course, that he isn’t), but she may or may not have any idea of whose child he really is. As far as Barristan, I don’t think he knows, but it’s not out of the realm of possibility. Obviously the three Kingsguard who were at the tower knew what was going on. The fact that Lyanna was pregnant couldn’t have been hidden from them, and persumably they would have been given some reason as to why they were to stay at the tower. But did Rhaegar tell anyone else what was up once he traveled back to King’s Landing or before his death on the Trident? There’s no way Jaime knows, we’ve been so far inside his head without any hint of that. But he was the rookie on the guard, Selmy had been around a lot longer. Rhaegar might have confided in him. I don’t think there’s been anything to point to it so far (and shouldn’t he have told Dany what he knew when he came clean with her about his identity if he did?), but it’s not impossible.
Anyway, back to Jon’s legitimacy and vows…
Really minor nitpick, but Aemon wasn’t a member of the Night’s Watch when he was offered the crown. He was a maester, but was still secretly offered the throne (by a great council? I can’t remember), but turned down the offer and then joined the Night’s Watch to further emphasize his refusal and to remove himself from the scene. But yeah, the maesters’ vows and the vows of the Night’s Watch are not that different. I didn’t remember that bit specifically about the High Septon, but if he could undo one, I’m sure he could undo the other. And I agree that a king could absolve a man from his vows if he was willing. For that matter, a king could make a bastard legitimate if he so chose. Damon Blackfyre was made legitimate by his father (some Targaryen king whose name I’ve forgotten). So neither Jon being a bastard (if he actually is) or his vows are an insurmountable problem.
But, at this point, I’m kinda reminded of Varys’s question to Tyrion in ACoK about the soldier and the priest, king and merchant (I think that’s how it goes). Where does the power lie? If Robb had made Jon legitimate and his heir, and absolved Jon’s Night’s Watch vows, as he discussed doing with Catelyn in ASoS, would that have been legit? Sure, to those who acknowledged the legitimacy of Robb’s reign. To the Lannister supporters, it would have been an illegal order by a rebel. Same with Stannis and Jon. Those who support Stannis or could be convinced to come to his cause would name Jon the Lord of Winterfell (if Jon had accepted it), but the Boltons and Freys and the south would have laughed at the idea.
So, I guess my point is, if Jon’s true parentage is revealed, and it comes down to who has a better claim, Jon or Dany, it’s not going to be an issue of studying the Westerosi legal code for precedent, IMO. It’s going to be an issue of: who has the power? Who can sway people to their side? And so on. A king’s orders only matter to those who accept that king. If, say, Stannis absolves Jon of his vows at some point, and then Jon finds out he’s the legitimate son of Rhaegar, and he puts forth his claim, Dany can just say that Stannis was never a legitimate king, so his orders don’t count.
Of course, I don’t think it’s ever going to come to that between them. If anything, if Dany finds out she has a nephew, I’d think she might welcome him, to continue that Targaryen bloodline, incest thing they had going. Which goes back to skeeving Kyla out. But then, when have I ever correctly figured out where this series is going.