To enable Amazon checkout go to System->Configuration->Amazon Import->General and set "Checkout on Amazon" to "Yes"

How to enable Checkout on Amazon from Magento - Magento Amazon Integration

Visitors from your Magento store would be transferred to Amazon cart. Add products to Magento cart and checkout button would transfer cart to Amazon cart. The affiliate code is kept in Amazon Cart for up to 90 days. You can get affiliate commissions even if your visitor waits 3 months to buy.

Magento Amazon integration

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