Nope, bow makers are bowyers. Arrow makers are fletchers. Archers didn't make their own bows.

England had a massive archery industry. Bowyers, for example, were craftsmen who went into apprenticeship for 7 years. There were guilds who controlled the business. There were price regulations for bows and bow staves (the raw product one might say) and fines for violations such as 'painting' a bow before it was certified (!) to be eligible for painting, working at night, etc.

Bowyers would often work by piecemeal. For example roughing out the shape of 100 bows at a time, and so on and on.

The only satisfactory source for this is probably the Great Warbow by Matthew Strickland and Robert Hardy.

Anything you want elaboration for?