They mention that there is a signal for viral reactivations from the vaccines, retrovirus in particular. I'm not sure if this also refers to mRNA vaccines only or to the Adenovirus vaccines as well.
The only case I found after a quick literature search is this one:
Herpes zoster emergence following mRNA COVID-19 vaccine (2021) [10.1002/jmv.27036]
There's some speculation that this might just have been coincidence, but I don't see any reason to believe so. The immune system is complex enough that any immunological regulation can cause this in a patient with an overall weak immune system.
We also have to consider that HZV is quite difficult to completely reactivate among all the herpes viruses. Other viruses like EBV might be reactivated more easily but not detected because nobody cares about other herpes viruses. The basic reativation mechanisms are always the same.