When we went to Kings Canyon we always stayed down in the canyon, I think there are four campgrounds in that area all following the river. I don't think any on them have reservations, all first come first serve. As long as it is not a holiday weekend I don't think you would have any problems.
It is a bit of a drive down into the canyon and slow going but not a problem for your trailer.
Check with the website I do know the Sequoia's have at least on campground you can make reservations. Believe area.
Also note that there are restrictions on your length going in from the south.
