How would I go about creating a customization to search for orders/customers by a product in their order
You can use the Moco Insight extension to do this. Easy to install with a 14 day trial so you can play around with it to see if it suits.
You may check this Order Management Magento tool. It lets you add a column with products names to the Orders Grid, and then sort orders by a product.
Also you may use these 2 free extensions: