Hi Wendy, Check out our site Imperialproductions.co.nz. We make to order and don't take payment up front, but instead use an app called Chargemelater which enables customers to assemble a shopping cart of items in our catalogue, "order" them, which creates a draft order in Shopify. Once we have made the product we invoice with shipping amounts and it turns into a normal Shopify order. The customer pays on the invoice and we then fulfil the product. It is a little untidy if you want the system to handle partial payment in a cart for a mix of made to order and in-stock items, so we just consider everything as zero inventory and if we happen to have it to hand, then the "manufacturing" time is short. Well eventually get to carrying inventory so may need to split the catalogue into a collection that triggers an order and one that can go straight to cart but we'll cross that bridge in the future.
Shopify's draft order functionality has all the raw ingredients for an effective make-to-order and pay on ready model but no-one has done a great job of bringing it to life the way we want. Drafts are "out of the loop" for reporting, notifications etc out of the box.
Hope that helps. Mark