Sam here with the Shopify team.
You can actually use the Partial Orders feature to hold the Products on Layaway until you receive the Payments and Fulfill the Order entirely if needed. This also works when Orders are placed Online in your Store and not just for the Point of Sale.
To do this, you would just want to make sure that under your Settings and then Checkout page, you have the Orders set to not be Fulfilled automatically.
Alternatively, you can definitely use an App such as Recurring Payments, Subscriptions & Invoicing by PayWhirl for Layaway and Subscription features as well.
Hope this helps!