When Wyman Manderly offered to build a fleet of warships for Robb, would those ships have been a Royal fleet where Manderly would have had influence on appointments, or would they have been a Manderly fleet, fighting for the Starks due to Wyman’s oaths of fealty? Did vassals need permission to build up their own military capabilities?

From the way that Wyman Manderly puts it…

Wyman Manderly had a great booming laugh. It was small wonder he could not sit a saddle; he looked as if he outweighed most horses. As windy as he was vast, he began by asking Winterfell to confirm the new customs officers he had appointed for White Harbor. The old ones had been holding back silver for King’s Landing rather than paying it over to the new King in the North. “King Robb needs his own coinage as well,” he declared, “and White Harbor is the very place to mint it.” He offered to take charge of the matter, as it please the king, and went from that to speak of how he had strengthened the port’s defenses, detailing the cost of every improvement.

In addition to a mint, Lord Manderly also proposed to build Robb a warfleet. “We have had no strength at sea for hundreds of years, since Brandon the Burner put the torch to his father’s ships. Grant me the gold and within the year I will float you sufficient galleys to take Dragonstone and King’s Landing both.”

…I would lean heavily towards the former. The customs officers serve the King in the North, and the king has the power to confirm them or not, but Wyman gets to appoint them (although he might have to pay for them as quid-pro-quo). Likewise, I’d assume that Wyman pays for the mint, gets to appoint the officials, Robb confirms them and keeps the right of seignorage, etc. 

So with the fleet, I would imagine that, especially if Robb is paying for them with royal funds as Wyman suggests, it’s a royal navy based out of White Harbor. Which means that there will be lots of offices and sinecures in this new royal navy for Manderlys and Manderly vassals, as per usual subject to the king’s approval. 

To answer your last question…it’s tricky. Under the normal rules of feudalism, military capabilities were limited by the terms of the feudal agreement – you get so much land, you agree to raise so many men, the number of men per unit of land is fairly standardized – and it was hard to alter that, because the vassals’ vassals know their rights in law and get pretty litigious about it.

It’s really more when you get to what’s known as “bastard feudalism” that things start to go off the rails. Under bastard feudalism, instead of relying on those feudal agreements to raise soldiers, you convert military service obligations into taxes paid in cash and then use the cash to put fighting men on the payroll, who wear your livery and are counted as members of your “affinity.” 

So now you have a system where noblemen can raise and maintain private military forces above and beyond their feudal rights – and the only limit to how many of these guys you have on the payroll is your ability to make payroll on the first of the month.

That’s what leads to “over-mighty vassals” starting private wars and making themselves extremely difficult to govern by their liege lords, because they might have more military manpower than their overlords. And that’s what eventually brings down heavy regulation where vassals did indeed need legal permission to have any military power whatsoever.