No they do not require tracking information for someone to buy an item. But they do indeed require a tracking number from the seller. If no tracking number, Ebay will not back the seller in anyway in any dispute, nor will paypal.
Ebay under sold items records all tracking numbers, and ship dates, and marks it appropriately when the item is shipped.
Also about 2 months ago Paypal got into the business of verifying shipments and delivered.. This was not something they used to do, but if you look at their policy updates, its right there.
In the past paypal did not get involved in the shipping aspect of transactions.When Ebay decided they were requiring tracking information to be eligible for seller protection, PayPal followed suit and requires it for any dispute now. They leave you alone until there is a problem.
They very much require tracking numbers, and you can also print your shipping label right from paypal or Ebay, and in your transaction ledger they provide a seller with a method to purchase a shipping label for cheaper than the post office rates. They ask you to buy a label, or enter the tracking number and mark it shipped.