Is it possible for a 13 year old teen to contract the herpes zoster virus with no history of the varicella virus?

Yes: In children, the rash is mild and complete resolution usually occurs within 1-2 weeks.

Terminology issue?: Varicella Zoster Virus (VZV) is the virus that causes chicken pox in acute infection. After being exposed to VZV, the virus can become dormant and later appear as shingles, or herpes zoster. For one to have shingles (herpes zoster), they must have been infected with VZV previously. The acute infection (chicken pox) can be very mild, maybe unnoticed, especially if the person was vaccinated.

Yes: In the pre-vaccine era it was well known that about 95% of adults with no hx of chickenpox had antibodies in their blood. This is evidence of prior infection that may have been unrecognized or called something else. Once the varicella germ is in your system it can come out of hibernation as active zoster. I've seen it in toddlers. You don't "catch" zoster, from others, it emerges from inside.