I live in Buffalo and have biked over the Bighorns lots of times.
There are plenty of places to get water.
Since you are heading east to west -
Campgrounds and lodges:
The Pines Lodge (Sold/closed)
Middle Fork Campground
South Fork Lodge
South Fork Campground
Crazy Woman Campground
Lost Cabin Campground
Meadowlark Lake Lodge
Deerhaven Lodge (Closed)
Boulder Park Campground
Tensleep Creek Campground
Not to mention that there are creeks every mile or so.
But you need to filter the water.
You should stop at the Forest Service office on the west side of Buffalo and verify if the water is on in the various campgrounds. Sometimes they have a problem with a well. Also, you can pick up a free travel map in B&W that gives you a more detailed view.
The climb out of Buffalo is steep for 10 miles to the overlook
then it has a series of seven roller coasters to Crazy Woman
then a gradual climb to Powder River Pass.
I suggest that you plan on staying at one of the mountain campgrounds to break up the climb.
South Fork is just across from the lodge with a lovely trail heading down the creek.
Crazy Woman is off the highway a half mile or so it is pretty quiet.
Of course, it's legal to camp anywhere on forest service land.
Sorry, but wildflower season is over