Just wanted to post here and mention that I did a little more digging after posting the other day and found the easiest way to do this is simply using a copy of the cart class that's in the session variables. If there is a better way to do this please feel free to correct me.
PHP Code:
# Retrieve the list of parts in our cart so we can get their product ID for shipping calculations
$ship_cart = $_SESSION['cart'];
$ship_products = $ship_cart->get_products();
# Create a variable to store our data for part ID's and qtys
$parts_to_ship = array();
foreach($ship_products as $key => $values){
$parts_to_ship[$values['product_id']] = $values['quantity'];
}