Okay, don't flame me if this seems like a stupid question. NH ATV rules state no firearms on your person while riding an ATV on public land. LEO Protection Act of 2004 says an LEO may carry a firearm nationally. If I chose to open carry on an ATV under the umbrella of the LEO Act, would there be a problem? Any NH LEO's around to give an opinion. I am thinking a Fed law would over rule the NH law.
The rules for OHRVs are the same as for motor vehicles: no loaded carry without a license. With a license, you can carry a loaded handgun (not long gun).
This is about hunting, or course.
I would not try claiming a LEOSA exemption, since LEOSA specifies "concealed".
215-A:20 Loaded Firearms Forbidden. – No person shall carry on an OHRV, or a trailer towed by same, any firearms unless said firearm is unloaded. This section shall not apply to law enforcement officers carrying firearms in the course of duty or to pistols carried under a permit issued pursuant to the authority of RSA 159.
AFAIK the laws that apply to vehicles apply to ATVs/OHRVs. That means, if you have a license to carry issued by the state of New Hampshire, you can carry concealed or technically openly with a license on an ATV/OHRV.