"little pieces of plastic and rubber that fell off the inside of the gas cap."

Is the cap venting?

I ran a thin wire thru the vent hole, and then blew it out with an air compressor, so I think it is venting. There had been an inner plastic thing with a little piece of foam rubber that disintegrated in the tank. All that stuff is gone now, but there is still a hard plastic disc inside the cap that hides the vent hole, but doesn't plug it up.

Does this engine have a fuel pump, or is it gravity feed?
If gravity feed, remove the fuel line to the carb and check for flow. (have something to catch the fuel until you get the line back on).
IF a pump, have a receptacle handy and crank the engine over a few times and check for flow. IF you have a vacuum/pressure gauge, check for pressure too.
IF good, check fuel flow TO the pump.

YOU VACUUMED THE DEBRIS OUT OF A GAS TANK! With what? I don't guess you ever heard of a vacuum cleaner turning into an air/fuel bomb from someone doing that? The vacuum cleaner motor shares some of the exhaust air that has passed through the vacuum collection canister. If the volatiles are concentrated enough and the sparks (which are always there) from the motor commutator and brushes ignite this air/fuel mixture there will be 2 explosions (at least 2). The first explosion happens in the motor compartment where the sparks are. Almost instantaneously, that small explosion will propagate back into the collection canister igniting a larger, more violent explosion. Further explosions and/or fire depend entirely on how much other flammable liquids are in close proximity to the vacuum cleaner. That's why your vacuum cleaner has a warning about not picking up flammable liquids. Most fire departments can tell a few tales of calls they have answered due to exploding vacuum cleaners caused by this. You wouldn't be the first, but it could be YOUR last.

Those motorized turkey basters are notorious for blowing up!

Real likely those bits of junk made their way down into the carb. They also have a way of getting bound up in the fuel pump and the curvey metal gas line to the carb. I would drop the bowl of the carb and do a basic clean of the carb bowl and body to start. At any fuel line connection drain some fuel out to try and get the debris out. You can check the flow as Bill had said while you are doing that. The check valve in the pumps on these do sometimes fail to- but strat with a good cleaning.

No fuel pump, so I'll check for flow. Line comes out of the tank to a little shutoff valve, then to a filter, and straight to the carb.

I had "turkey-bastered" all the liquid out, then mopped it out with a piece of chamois and let it sit in the garage over night with the cap off. Then I blew air into the tank for several minutes with a compressor. I was probably wrong, but I felt like all the fumes were out of the tank before I vacuumed it. That said, I'll count myself lucky and not do it again!

Needless to say, I hope it turns out to be a blockage in the fuel line. I'm not sure I'm up to rebuilding the carb ,,, I've disassembled lots of stuff in my time, and always manage to lose my way.

For those of us who sometimes tend to "lose their way" between dis-assembly and re-assembly, I will mention again, "Use a digital camera and take lots of pictures of the operation as you go along. For those folks who would say they "don't like all them new fangled ideas and inventions": What are you doing sitting at a computer? :^) But seriously, if you have a digital camera you can take enough pics from all angles to make reassembly a piece of cake. If you have ever felt that a certain service manual (or whatever), could have had a few more pictures for clarity, well, there you go.