What is wrong with my friend's vehicle (details inside)?

Big vehicles are susceptible to even gentle winds. So , it’s possible that you just aren’t used to being in that big of a vehicle.
The shocks could certainly be soft.
If the wobbling got worse at higher speed, it could be tire issues, or a part in the hub, or axle.

@MrGrimm888 thanks, I don’t believe it is a wind issue. It can happen at low speeds it is just more pronounced the faster he is going. Most of the trip was done in the 65 – 75 mph range and thinking about it I seem to recall that there was something, a bump or an uneven road, that made the vehicle start to sway. It didn’t just happen, there was a small and what I would consider insignificant boost that make it noticeable.

I was in a vehicle which did like that. It turned out there were three busted motor mounts.
Lots of stuff can go unnoticed. Sometimes the culprit turns out to be something you would have never thought.

It could also be something as simple as low air pressure in the rear tires.

I’d say you’re right on with shocks and possibly add alignment to the list.

The alignment is really easy to have checked, some places do it for free*, so I’d start there. I’d never had it done on my first umpteen cars, and finally got talked into it when doing some brakes, it made a huge difference.

* – actually free, like they’ll check and then give you a report, which you can drive away with

Check the bushings in the anti-sway bars and the action of the shocks. The shocks should show no signs of leaking oil and when the vehicle is bounced aggressively, should not bounce more than twice. (Get some body weight on it, and make it bounce. Jump off and count the residual bounces.)