Is there a way to make my own chalkboard paint without using unsanded grout?
You need unsanded grout, paint, and chalk.  

Mix your unsanded grout.  

Then add the unsanded grout to your paint.  The ratio is about 1 unsanded grout to 10 paint.  

Then you have to mix really well.  If hand mixing this could be a good 10-20 minutes.  

Paint section.

Let it dry a day.

Rub chalk all over the paint very thickly.  Let it sit for about a half hour.  Wipe of chalk with rag or eraser - nothing wet.

Should be good to use.