It's important to distinguish between the Domestic African Slave "Trade" and the Trans-Atlantic Slave Trade.
Long before the Portuguese arrived, domestic slavery existed within Africa, and there was a market in buying and selling humans. Black slaves had been traded by Arabs, and exported to the Mediterranean as well as the Near East. Following this pattern, the Portuguese started exporting slaves to Europe; when the English began in the slave trade the more virulent and ugly form of Trans-Atlantic trade that we understand came into being. However, this was accomplished only by the acceptance of the trade by African rulers -- most slaves taken from Africa were sold by African leaders who grew rich in the business. Read more at the link:
Yes, the slave trade did depend on the people of West Africa.
The slaves who were brought to the Americas were taken from West Africa since that is the part of Africa nearest to the Americas. Taking these slaves required the cooperation of at least some West Africans. The slave traders were not strong enough to simply go in and kidnap slaves themselves. Instead, they bought slaves from the people of the coastal West African states. These people captured the slaves and sold them to the traders. In this way, the slave trade did depend on West African people.