Is It Normal to Get Facial Sores After a Chemical Peel?

A properly performed chemical peel should not result in facial sores or open ulcers. A herpetic infection is a possibility but this would initially appear as a cluster of small blisters. For most deeper chemcial peels patients are treated with an antiviral agent to help prevent this complication. If a chemical peels goes down too deep in an area is can result in a deep skin injury. There are many variables involved including how your skin was initially prepared, what type of peel was done and what agents were used. You should report this finding to your physician as soon as possible to prevent any possible scarring.

I'm not exactly sure what you mean by sores. You should contact your physician as soon as possible though.
Chemical peels and laser resurfacing can cause a herpes flare up. (~50% of teenagers have been exposed to HSV-1 and 80% of adults have been exposed to HSV-1).
The herpes virus lies latent or dormant in our nerve endings. More often than not it does not spread and does not flare up. So, even if you've never had a herpes flare, you may still have been exposed to the virus.
Peels injure the skin and can lead to a flare up, i.e. a herpes outbreak. This is something that can be treated medically, but should be done so quickly.