Who this reaches
On this page11
The verbs are the scope
Possess, offer for sale, sell, offer to barter, barter, offer to purchase, purchase, deliver for shipment, ship, export, import, transport, carry. Almost every way a fly moves is named.
Which catches the hobby as well as the trade
Swapping flies with another tyer is bartering. Posting a box to a friend is shipping. Taking them on a trip is carrying.
Commercial tyers are squarely in it
Selling flies is the most obvious of the listed acts, and a tyer selling through a shop or online is doing several of them at once.
And material suppliers too
Offering for sale and purchasing are both listed. A shop selling feather is inside the same sentence as the tyer buying it.
The clause about products is what makes this different
A finished fly is not a bird and not obviously a part. It is reached as a "product, whether or not manufactured, which consists, or is composed in whole or part" of one.
Whether or not manufactured
Four words that close the gap somebody would otherwise argue. Working the feather into something else does not take it outside the Act.
Unless permitted by regulation
The prohibition opens with that qualifier, and permits exist for specified purposes. Whether any applies to a given situation is a question for the Fish and Wildlife Service rather than for this page.
Inherited material is the hard case
Old tying collections contain feathers from an era with different norms. Possession is in the list of prohibited acts, which is why this comes up and why it needs real advice rather than a catalog page.
This property does not give that advice
It names the clause and the species list and stops. Guessing at which feather in an inherited cape is lawful would be exactly the kind of invented answer this catalog avoids.
For new material the answer is simpler
Buy from suppliers selling domesticated poultry bred for tying, which is what the trade overwhelmingly is. That is the next section.