Engine not cranking over - bad starter?

Please help, at a total loss. Have a 3.8 liter V6 with 161,000 miles. I have spent a lot of money in diagnostic repair bills.

Problem: After the van has been driven for a while and turned off, sometimes, with no warning, when I turn the key to start the engine nothing happens, the engine will not turn over. All lights (inside and outside) work, the van will not make a sound.

I have had the vehicle towed, several times. As usual, when the mechanics finally get to it, the van starts right up. The pattern seems to be; van will initially start, then unexpectedly, days or weeks later after the van is driven and turned off, the van will not start. Or, if it does start, it will sputter and shut off.
After a time or two, it will not start back up.

However, after letting the van rest an hour or so, the van will start right up and run again until the problem repeats itself. Could something be loose causing the problem or something more?

Diagnostics have been completed and no codes have been found by the mechanic, because, the van works fine when by the time the mechanic gets it. Anything you can do to help would be greatly appreciated. I do not want my wife and children to be stranded again.