Are there different thickness heads/nuts for hex bolts?
You probably don't want to do that anyway! Think carefully of what your request would mean.

A 3/8-16 thread means that the width of the bolt is essentially 3/8 inch, but what matters here is the 16. That tells us the thread pitch is 16 threads per inch. The nut being about 3/16 inches thick means that the bolt and the nut mate on three threads. A thinner nut would have less thread contact, so less strength.

If you really need a thinner nut, then it will be easiest to use a smaller diameter bolt. They have finer threads, so the nuts are also correspondingly thinner. Of course, I don't know why it is that you have chosen a 3/8-16 bolt. You may also choose a different thread pitch for your bolt. The 3/8-16 thread pitch is the standard (coarse) thread that you will find in nearly all home stores. However a more diligent search will find bolts and nuts with a 3/8-24 or even a 3/8-32 thread. They may have nuts in a different thickness.