How do I make sure that Infusionsoft is not sending out receipts for orders being placed in my external shopping cart?

You need to check in two places.

1) In the E-Commerce section, click on E-Commerce Setup. Under Payments, click on receipts and make sure the check box is not checked.

2) In the E-Commerce section, click on Settings. In the left navigation, click on Orders. Make sure the radio button next to "Email invoice upon successful payment" is not checked.