Yes, you can definitely have soreness after throwing up.
Your rib muscles, as well as your stomach muscles, contract violently when you are throwing up. The rib pain is known as Costochondritis, which is basically a muscle strain in your rib muscles. It's more common on the left side, though I'm not sure why, and it can also occur after a lot of coughing.
It's nothing serious, but it is gonna hurt for a week or two. Just try to take it easy, and taking some ibuprofen will help with the pain and swelling.