Custom product form or Checkout appointment scheduler

This is difficult. There may be options but many will be unsuitable.

You could add a date picker to the product form with restrictions on the available dates, but they would need to complete it for each product. If customers can purchase more than one product, it won't be suitable for you.

You cannot add a date picker or restrictions to the checkout page because it is completely locked down because of the sensitivity of payment information.

You cannot control how many products are picked up on one day either.