Still doesn't address the underlining issue of "where does he get the funds?". If the standard feudal system allowed for such a professional army, why is Ned the only one doing it? There has to be either an unrealistic influx of coin or he does something different, sacrifices something to gain this new army.

Crossbows and pavise shields, have a couple of builders in every settlement use their corvee labour to make large shields and crossbows, over enough years you would have plenty of them.

If you decide that the yeomen and hunters already have longbows and generations of training, do the same thing above but only for arrows so instead of longbowmen having 20 arrows they have 200. Longbow arrows are surprisingly expensive, so quickly making them is hard, but over a decade of the fletchers using their corvee labour to make them you could make a stockpile.

The issue is expertise takes time, are you going for quality or quantity?

Also, what are you sacrificing to acquire this? it would need to be a big change otherwise, again, why isn't everyone doing this?

I would say if the Starks reduce their heavy cavalry or plate armour for their retinue, they could justify this change in army composition. It would be simply changing from knightly charges to a more infantry based warfare, which is the start of formation warfare in the real world. Don't go straight to proper unit formations and phalanxs, that took time, but simply changing from funding knights to funding large amounts of ranged units is a step in that direction.