This is not much different than the way we travel most of the time. There are three things you will need that take planning:
Provisions - Walmart is easy
Potable water - Many places have, but you may have to look
Sewer dump - This is tougher. At Yellowstone, you should be able to get to the dump and potable at Fishing Bridge RV.
There are websites that can lead you to all of the above, but if you are not carrying internet access, you will have to find it to stay on track.
There is, however, I small issue here. Missoula to West Yellowstone is 300 miles (+/-) and it is not all interstate. Both you and the kids will need breaks. That will be one LONG day.
I will suggest that you stop in Boseman at the Museum of the Rockies. D5 will get some, but most of it will be lost for the S2. But we found in later years that very the very young and not yet communicating well can absorb a lot more than you expect.
There is no problem with overnight parking at places that allow. We do that all the time, even when we have grands along. Safety in no issue outside of urban areas.
About Yellowstone, The primary roads are a north and a south loop. Get the maps. A loop a day is about what you will be able to manage. Do not plan on see much from the loops. Be ready to pack out and hike a lot. A Fast Survey of Yellowstone takes about three days. Get everything you can on line before departure.
If you don't dive into some trip planning site tonight, this excursion is bound to end up a disappointment. Get on to all of them. Many use RV Trip Wizard, but I recently found Furkot and like it except that it is online and we frequently have no access.
Planning here will be your key, but be flexible and expect your carefully assemble plan to be at least off if not trash by the third morning.
Take the time to enjoy the children and the sights. (You do know that you get to teach the daughter to drive shortly after you get home and then she will be gone.....
Matt