I'd do either of these options:
1. Get Yakima or Thule rack setups to fit the number of bikes you need to carry and put them on the van. This is what you see a lot of bike tour companies do as it offers good flexibility. Most rental vans are Ford E-series, which take standard rain-gutter mount towers (like the venerable Yakima 1A tower, which can be found on Ebay and CL for cheap) for the racks. If you are sure that you will be renting an E-van (Ford E150, E250, E350, or Clubwagon) this is probably the easiest and most secure way to go, but it could get expensive. And loading on top of a high van is not for the faint of heart as you'll have to climb up on top and have somebody hand up the bikes. Also, some rental places are moving to the new-style Ford Transit vans. I'm not sure if they still have raingutters like the E-vans do. As far as the rental company allowing it, how would they know (unless you put it on in the parking lot or something)? Raingutter racks go on and off quickly and easily. As an aside, if you aren't familiar with driving loaded full-size vans, be careful. When they are loaded with people and gear they are very heavy and require extra skill (I have a Ford E350 campervan and am very familiar with them). Don't overload the van! Basically a van full of people is at max capacity, even without carrying gear (10 people @ 150 pounds = 1500 pounds or 3/4 ton).
2. Rent an enclosed U-Haul trailer. This assumes that your rental vehicle has a hitch on it. Not all do due to liability issues. To protect the bikes you can either build basic wood racking inside (you'll need to get the trailer well ahead of time and be handy to do this) or simply use foam sheets between the bikes. Since UHaul trailers are standard, you could rent one ahead of time for a day or two to fit and make your racks inside, store them until your trip, and then you are ready to go.
3. I'd also consider renting a separate U-Haul truck for the bikes and gear if you have enough drivers to support a second vehicle. If it's a supported tour, presumably you'll be hauling people's luggage around too and you need the extra room anyway. For this option, you could build some basic wood racks as noted above. Or simply use moving blankets, cardboard bike boxes, etc., to keep the bikes separated. Assuming you also have a passenger van, get a basic rack that hangs off the rear doors for the occasional side-of-road sag service. If you need to transport the whole group (bad weather, etc.) then use the box truck.
