During the checkout process, the order is not created- it is only created after the final step of the process. At this point, you are only working with the shopping cart (ECommerceContext.CurrentShoppingCart) and you can only work with fields of that object. Fortunately, there are two helpful mechanics that you can use to accomplish what you're looking for.
Firstly, there is a pre-defined ShoppingCartCustomData field that you can use to add custom data to carts, and this data is automatically transferred to OrderCustomData when the order is created. You can add items to this field by referencing a key like a Dictionary: ShoppingCartCustomData["MyField"] = 123.
If you don't want to use that field and really want a dedicated custom field for your data, you can create the field in both ShoppingCartInfo and OrderInfo. If you name the field exactly the same, the data from the cart will be transferred to the order, just as above with ShoppingCartCustomData. So, you need to create 2 fields named "MyField" (for example).
All this said, the best way to modify the cart data during checkout is using a custom web part- I'm not sure what your data is and where it comes from, but generally some user-interaction is required (e.g. a form) and the web part takes the input and saves it to ECommerceContext.CurrentShoppingCart.