Is it normal for a company to ask for the last four digits of your credit card after you have placed an order?

It's moderately normal.

Most companies have ordering systems that store credit card numbers that have been used to purchase items from that company. (This is not for any nefarious reason; in order to issue a refund, for example, the merchant must provide the full credit card number as well as the date and the exact amount of the transaction.) They're (hopefully) stored in an encrypted form, so the people using the system cannot see the full credit card number, but they usually can tell what the last four digits were. Asking you for the last four digits of the card lets them avoid typing the whole thing in again, since it's very unlikely that the same customer will have two cards on file with the same last four digits. They may also ask for the CVVC (this is the 3 or 4 digit number on the back of the card... their agreement with the credit card processing company forbids them from storing this number).

You don't HAVE to have a CVVC to run a credit card, but you do get a discount on the transaction fee if you do, so it's policy at most places that take credit cards over the internet or by phone to get it.