This is what analytical chemists do when presented with any analysis problem: find a way to separate the material of interest from the background and then use an appropriate detector to measure the material.
Separation
Separating the chips from the vaccine is almost trivial, albeit tedious. The sample is placed in a centrifuge and spun. This will push the chips into the base of the tubes and the liquid is poured off. The chips will remain embedded in the bottom of the tube.
If the experimenter finds it convenient, the tube can be reloaded with another sample and spun again. This would collect two batches of chips in the tube. This could be repeated a number of times, populating the bottom of the tube with as many chips as desired.
Now, this likely would make it necessary to design a sample tube, particularly for this purpose. The inside bottom needs to be flat and the outside bottom needs to fit the centrifuge, which is likely to be round. The bottom should also detach from the cylindrical wall of the tube. But any competent machinist can design and fabricate it — it only requires a bottom that can be unscrewed from the tube. Simple.
As for the lab equipment — Amazon lists multiple centrifuges advertised to produce 5000-6000G for roughly 300-400 dollars. God only knows what a full-fledged scientific supply house could provide. 5000G must certainly be plenty. I shudder to think what a real laboratory-grade device could produce.
As for the vaccine samples themselves — I can’t imagine this would be hard. When the vaccines were first released, the early batches didn’t get to pharmacies before they were nearly expired. Some early batches were discarded. Ah, well, so it goes. At any rate, each vial had five doses, which means a minimum of five chips. Five or ten vials yield 25-50 chips. Surely one chip out of 25 or 50 could be found.
Analysis
The next step is to use a scanning electron microscope (SEM) to view the tube. Unscrew the bottom of the sample tube, mount it in the microscope, and look for the chips.
SEMs are expensive, and SEM operators are rare. Baseline SEM models will provide a resolution down to 10nm. High-end models will approach 1nm.
Transmission electron microscopes will do better than SEMs, but, so what? It will turn out that 10nm is more than sufficient to limit the chips to a size that makes them incompetent.
Access to SEMs is a bit difficult. But in a worst-case scenario, it should be easy to set up some sort of GoFundMe project to collect enough money to buy an SEM and hire some experienced analysts for weekend work at premium pay.
In principle, it would only require four samples. A blank, a spike, the sample, and a spiked sample. The blank is what it sounds like — uncontaminated water is handled just like the sample. The blank is done to show that the experiment won’t produce false positives. The spike is water plus some deliberately added nanoparticles; this shows that the experiment can detect particles if they are there. The sample is the vaccine, of course, and the results of that experiment show whether the chips are there. The spiked sample, in this case, is not strictly necessary, but in the interest of good form, it might as well be done. The spiked sample shows whether there are subtle interactions between the sample and the nanoparticles; the results should be consistent with the other experiments. About the only way the spiked sample experiment would matter is if the result was that the nanoparticles disappeared.
It’s easier to buy nanoparticles (for the spikes) than I guessed, though it’s hard to get amounts of less than 20g or so. That amounts to a nearly infinite supply compared to what would be needed.
The analysis, to sum up, might be expensive but nothing is challenging about it. I will warn you that attempting to get an analytical lab to do this on contract is likely to fail — such a small number of samples are needed that hardly any of them will want to take on the project.