I used to live in Alaska and have been back to Denali 3 times in the last 5 years. We will go back next summer. We love it there.
Some of the information above is incorrect or misplaced. For example, the Tundra Wilderness Tour is not the longest tour into thepark. It goes to Mile 63 on the park roadd ( Stony Hill overlook) and returns to the start, taking 7-8 hours. The longest tour is the Kantishna Experience which goes all the way to Kantishna, basically the end of the road. This one is 12 hours. There is also a shorter tour, the Natural History Tour, which is not recommended as it does not go far enough into the park.
You can read about the narrated tours here:
The "destination experts" on the TripAdvisor Alaska forum, who are very knowledgeable and helpful, will uniformly advise you to skip the Tundra Wilderness Tour and take the regular park shuttle bus instead. It is much less expensive ( saving almost $100 per adult, you can go as far as you like ( Eielson Visitor Center at Mile 66 is recommended), and you can use it as a hop-on, hop-off tour, getting off where you like for a short hike or time to take photographs. And while these are technically shuttle buses not guided tours, some of the drivers do offer a lively narration.
The shuttle buses do stop when wildlife is spotted so you can see it.
With nine of you, renting a van or two cars isn't the only reasonable way to go. The train would be horribly expensive, and the timing of arrival and departure is not convenient to your plans. Trains coming from Anchorage arrive late in the afternoon (around 4) and the southbound train the following day departs back to Anchorage around 12:30, leaving you no time for a tour into the park. This is why the cruisetours using the train have their clients continue on up to Fairbanks after their Wilderness Tour. ( Way back when I worked in Denali National Park, then called McKinley NP, they used to start the Tundra Tour at 4:00 am so people would be back by noon to get the southbound train to Anchorage. We heard lots of complaints about that but it was the cruise company's schedule, not the hotel's).
Most importantly, if you take the train you would have great difficulty getting around from the train station to your lodging, from lodging to restaurant, and from there back to the park for your shuttle or tour. So I suggest you stick with the car rental idea.
Finding lodging for 9 people is going to be a challenge. The area right by the park entrance, known as "Glitter Gulch" for its pricy hotels, is usually pretty well booked up in August, especially the more affordable rooms. The favored lodges down the Parks Highway such as McKinley Creekside Cabins (our personal favorite) book up early as well. Carlo Creek Lodge across the road is less popular and more likely to have something.
I do not actually know of any VRBO or Air BnB rentals, but there might well be some in Healy, ten miles north of the park entrance. There are some nice motels, cabin resorts, and a hostel there as well.
I do know of one place with cabins and condos right near the park entrance that might work; i will go find the link and be right back!
But first I must agree with all the comments above that you cannot count on seeing Denali. I have seen the odds given as around 30% on any given day. Low enough, in any case, that when word went around the employees' area that "the Mountain is out" we would all jump into cars and drive the 25 miles or whatever it is to the first viewpoint, just to see it. ( This was in 1971, before the road access was restricted).
On our last two August visits, staying 3 nights at a wilderness lodge deep in the park, with a direct full-on view of Denali (when not obscured by clouds), we enjoyed views of the mountain most of the time during our 2012 visit. In 2015, visiting the same three days in August, we never saw it at all.