How can I rent a high clearance vehicle in the US?

You're not the first to want this. This means a couple things. First, the national car rentals are savvy to the requirements of driving in the area, and can counsel you, and may have offerings for your needs.

Second, you are a newbie to this, and they see a dozen people a day who want to off-road but not pay. They know how to head them off or find artful ways to rob them blind. Off-road specialty rentals charge more because their genuine risk is more. Which makes it plain fraud to use false pretenses to get street-driving prices, and gives sly off-roaders no defense against a $1000 cleaning charge or $2000 for "suspension repairs". You don't want to get caught up in that meat grinder.

The answer is, contact the auto rental's local office, who knows the territory, and tell them what you plan to use the car for. Ask for their best rate under those conditions. Get a usage waiver in writing, because words mean nothing. Then check with your own insurance too.

I know you're not off-roading, but a 406MHz ELT is a really, really, really good thing to have. Hokey things like "SPOT" are not the same.

You will not be able to specifically reserve a High Clearance vehicle since that is not an feature most agencies surface. An alternative is of course specialty agencies.

However, you can ultimately choose a High Clearance vehicle by working the system just a bit. Here's some tips:

1. High Clearance is >=8 inches (20 cm) and many SUVs (Sport Utility Vehicles) meet this.
2. Book a mid-sized or higher SUV but prepare to be flexible on features and price.
3. Bing and Google are your friend. When you get to the counter and learn the exactly vehicle, search it with 'ground clearance'. It's very easy to ask for something else.* The Agent won't ask what you searching, and if they do, just say 'safety ratings'.
4. You can pre-search the SUVs a particular agency features to save time.

There's lots of YouTube videos on the Toroweap Overlook road, many in apparently factory SUV's.

This site describes the road as 'easy to moderate' and 'suitable for passenger cars': AZ Offroading

*For clarity, this is not at all unusual, to request a different vehicle than the one assigned. Frequent renters do this to get specific features or even cars they prefer.

An alternative option, especially if you're from the US, is to see if you can find a vehicle for sale on the likes of Craigslist. If you can buy a vehicle and drive it for a couple of days, then you can probably sell it for less than it would cost you to rent one. Of course then you still have to deal with the hassle of selling it again, depending on your time commitments, maybe that's worth it.

We had (almost) exactly the same plan last year when we were camping on Point Sublime. We had booked the largest SUV to drive around the states (even with full insurance), but didn't want to take the risk of getting it stuck in the mud in the middle of nowhere. So we additionally reserved a dedicated off-road jeep locally for two days (all-in cost about $200 for six people).

We got lucky and actually got the biggest car in the lot (Chevrolet Suburban, 4-wheel drive, higher than normal ground clearance). We kept an eye on the weather, checked with a local park ranger, which assured us the road was "in really good conditions" and decided to cancel the jeep and go for it with the Chevy.

Everyone has his own definition of "Good conditions"; we made it, but it was still a difficult ride. If we didn't have that much ground clearance, we would have scraped the bottom of the car over the rocky road or got stuck.

So I'd still advise to get the proper vehicle for the job and definitely inform with the park rangers about road conditions.