A bit later in 1545 Roger Ascham's Toxophilus discusses bowyers and fletchers as very separate professions. And mentions a shop to buy bows from.

Also in the beginning he emphasizes that he's not about to tell archers how to build their own arrows, but how to shoot, so fletchers should not be angry about the book:

Yet, as I said before, no honest fletcher will be angry with me, seeing I do not teach how to make a shaft, which belongeth only to a good fletcher, but to know and handle a shaft, which belongeth to an archer.

Later he also talks about stringers, i.e. string makers, as a yet another separate profession.

So it would seem to be you'd have to shop around. And that actually makes sense because you will use dozens of arrows in the lifetime of that bow you just bought.

Edit: Typoed the year, fixed.

Expanding on OP's question a bit, how separate were these professions in, say, 1200-1250 CE England? Would making bows and making arrows be separate professions?

Suppose a small settlement, like a knight's fee with a population of 200 people. Would the knight rely on the same village blacksmith that shoes his horse and makes farm tools to produce arms and armor? Would professional armorers only appear in larger centers of power? Or would the resident metalworker be a professional armorer, who fills in his time between wars doing simpler blacksmith work?