Short answer:
The app will work to take orders, but inventory for each product within the bundle will not get automatically decremented until the payment has been processed. Depending on how quickly you can collect on delivery, this may mean your inventory won't be automatically decremented for a few days.
Long answer:
The way our app works is by creating a custom product for each bundle created, and then when the payment is processed, this custom product is removed, and Bundle Builder has edits the order to add each variant separately. At that point, when the individual variant is added separately, the inventory is decremented in Shopify.
If you use COD, the step of processing payment is asynchronous, the inventory is not going to be decremented in a timely manner. As a merchant, this could allow you to oversell your inventory.