Can you camp along the Brule river?

Yes

I was under the understanding you could camp in the state forest as long as you were 100 yards off the river and didn't cut anything down or cause harm to vegetation.

There are only two "river camping" sites on the river, both at Bois Brule Campground. They are first come, first served and nearly always taken since you are allowed to access them from the parking lot. Thus, the answer is effectively "no."

There is a campground that's located on the Brule river (brule river campground). A few of the sites are on the river and some are not. Great place to camp!