Can I have my own products and Amazon products on my site?

You can have both Amazon products and your own normal WooCommerce products on your site.

How does this work?

If a visitor adds one of your own products and some Amazon products on your cart, upon the checkout process a new window will open in a new tab with the Amazon cart containing the Amazon products while on your page, your products will follow the normal WooCommerce check-out process.

Just keep in mind that most web-browsers will block the Amazon page as it will see it as a pop-up.

Also turn off the pop up blocker and add your domain name to the exceptions list.