The catalog in New Leaf doesn't unlock until you upgrade the Nookling Junction to T&T Mart. This will unlock the catalog machine, which functions in the same way as previous entries in the series did, letting you order (most) items that you've previously collected/sold.
In order to upgrade Nookling Junction to T&T Mart, you must meet the following requirements:
- Complete the 10,000 Bell house down payment
- Spend 12,000 bells at Nookling Junction
- Either 7 or 10 days need to have passed since town creation.
The Animal Crossing Wikia states 7, but Nookipedia states 10. Other sources I can find also state 10, but it'll be difficult to verify whether or not this has changed from the Japanese version until sufficient time has passed from the release date.