This includes a feature that allows shoppers to request to be notified when a product becomes "in-stock". You could potentially use this as the basis for addressing the need that you described. It would give you a way to collect the e-mail addresses of shoppers that want the product, and a way to reach out letting them know when they can come back and make their purchase. You would, potentially, want to change up some of the wording/labels based on your description.
There are pre-order extensions for Magento, but these generally expect that you'll know what the price will be for the product is, even if it won't be shipped immediately. Whether it's a video game or a pair of limited-edition sneakers, the product price is established, and the customer purchases knowing that the item isn't in-stock yet, and perhaps their card won't be charged, or they'll only be charged a deposit, in the meantime.
There are extensions that will more closely meet the specs that you suggested, but they're typically built for quote systems. They allow shoppers to request a product, and a website admin to then provide an actionable quote that the shopper can use to checkout. So shoppers would request the item, and you'd send them an order to consider which they'd hopefully decide to proceed with and pay for. Here are some examples: