UL certification and laser tag equipment
It's not just the charging bricks. If you have been making laser equipment it's likely you're familiar with these two laser product safety standards:


FDA/CDRH 21.CFR within the U.S.
IEC 60825-1 outside the U.S.


It's not a legal requirement that UL certifies your lasers, but they must be compliant with 21.CFR and IEC 60825-1 for the appropriate power class.

UL offers a service to certify your product for compliance with these standards. More here: https://ctech.ul.com/en/services/safety/optical-radiation-services/

It is likely that your customer has the UL requirement on the gear as a condition of their insurance, which is why they're asking you for it.

As far as the suits and targets, if they don't themselves emit laser energy, not sure where UL would come in to play. The goggles, maybe. UL could certainly tell you.