The origins myth...

The origins myth in the promotional products industry has it that the very first promo item was a seed bag printed with the name of the seed supplier.
Not to be outdone, another local seed supplier made his bags into book covers for the local school.

Who knows?

Now the vast majority of promotional items are made in Asia with most originating in China.

Here's how the industry is set up.

Factories in Asia produce items under contract from suppliers here in the US. Most of these factories are contracted but a few of the largest suppliers have their own factories.

The suppliers here in the US import the goods blank...with no logos.
The suppliers warehouse the blank items.
Distributors place their order with the supplier, providing the end-user's logo.

The suppliers put the end-user's logo on the items and ship the completed order to the end-user.

Distributors are the intermediaries between the end-user and the suppliers. They're the promotional products vendors that the end-user works with. Distributors coordinate the order flow, handle all the back and forth and ultimately make sure that the end-user gets the product they want on time.

So, what did we learn?

Factories in Asia (mostly China) manufacture blank promotional items.
They ship them in bulk to suppliers here in the US
The suppliers warehouse the blank items. They then imprint them when orders come in.
Distributors coordinate the order flow between the end-user and the supplier.