Why else would they be called "mini" vans?
We use our Ford Windstar with a homemade safety barrier. I would NOT transport a horse that I didn't know well, as it does have windows. Headroom is 48".
Before we had minis, we owned the ideal horse hauler: a GMC Safari mini cargo van (same as a Chevy Astro). These are built on a truck frame, not a car frame like most minivans, with a boxy shape and high roof. Ours had no windows, had rubber mats throughout and a moon roof in back for ventilation...
We plan to purchase either a full-sized cargo van or a used ambulance to convert into a dedicated horse hauler. Ambulances sell for very little, have a low center of gravity, great suspension, lots of storage and a door from the cab. Sadly, they don't come with the lights and siren.
Either way, I REALLY like having my horse in the vehicle where I can keep an eye on him and know the temperature, etc. Mingus likes having his own stereo controls...