The problem is somewhere between the ignition switch and the starter. The dash lights up, stays lit up when I turn to start position, but nothing happens. No solenoid click, zilch, even with a jump. What I've found so far suggests a starter relay that's part of the security system. But my car doesn't appear to have one. All the cars discussed on the pages I could find, have the key with the resistor in it. Mine doesn't have that, nor does it have anything else resembling a security system. So my question: is it likely that this car does have that relay? And if so, where the heck is it? I haven't been able to find a location diagram, either.

Thanks!

I don't see a Starter Relay for your Buick. Be sure to check/replace the Neutral Safety Switch, which is between the ignition switch and the starter motor. If there's a problem with it, it will prevent the engine from starting. I would perform an ignition bypass test, where you connect a 12AWG wire, or remote starter switch, from the positive battery cable to the "S" terminal on the starter motor solenoid. If the starter motor cranks, the problem is with the ignition switch, neutral safety switch, wiring, or starter motor relay (which your car doesn't have).

You may have a battery that is on the verge of failing. GM side-post batteries have a bad habit of the terminal on them pulling loose. Look at the battery and see if there is any evidence of fluid coming from either battery terminal. Gently pull on the battery cables to see if the battery terminals are secure. Eye protection and gloves are suggested for your safety.

Turns out it was the starter itself.

I just replaced a starter in a 2006 Kia Sedona that had the same symptoms. No click, nothing. I pulled the starter shield off and it looked like it had been replaced recently as the Kia label was still pretty shiny. I started to fear that I was wasting my time as it wasn't easy to take out (had to remove the trans mount and O2 sensor)...I figured the likelihood of it being bad was slim. Anyway, I got it out and it rattled when I shook it and crap came out of it. I put the new one in and it worked. Got lucky.

The symptoms you described aren't very common for a bad starter but they do happen.