Even before the pandemic the equipment needed to do PCR testing was incredibly common. It isn’t just research labs or medical testing facilities that have the machines. DNA evidence being presented in court is the result of a PCR test. The ancestry testing companies are using PCR. You can find pages full of used equipment on eBay.
I can’t answer your main question on how many people are looking for viral inclusions into human DNA. That isn't information I would even know where to get.
I know some labs are looking for that.
As I understand it, the new "vaccine" works by including a likely antigen prominent on the surface of the virus into another, a carrier with relatively safe known characteristics, virus which transports the one Covid 19 gene, say for the "spike'" known to be far and away the most "visible" or accessible binding site of Covid 19, either in the form of an RNA or a DNA sequence, which is then expressed in the host and released from the infected cell.
Since this is only one gene of many necessary to the Covid virus, it is not itself a potential disease vector for Covid, but it causes the production of antibodies and full cellular immunity to the Covid virus. This normally causes, and is believed in this instance,to confer long-lasting immunity to the virus. The main concern is variability or changes in the virus over time that might mean people no longer have immunity to the current variant of Covid.
Some critics of various levels of understanding have gone "Holy Hell" this is gonna create a mutant Human. Many very competent scientists argue that these vector techniques are processed fairly quickly, catabolized and eliminated the system before any possible "inclusion" event can happen. This would make an inclusion event rare, as is believed to be the case in nature generally for such cellular viral attacks or contacts.
Even host mRNA and mDNA products are rapidly catabolized as their purpose of designed function is generally short-lived. Rapid turnover is one form of regulation of systemic active agents. So there is always a level of presence of enzymes that do that.
There is nevertheless one report of evidence for inclusion of the Covid viral strand from the vaccine, or possibly even from the virus itself in natural infection. This is an important thing to study. It appears to be one of perhaps several reasons why asymptomatic people are testing positive for Covid 19, so so-call "False Positive" problem. In such cases, even with a sustained level of viral protein being made, there is a continuing immunity. Such a pattern could trigger some kind of autoimmune process. We have to study this kind of thing.
It is at a very high confidence level something that is happening in mitochondria not in the nuclear DNA. Few "vectors" get into the nucleus.... well..... some do, not most.
Just for the curious, we humans have benefited from, or benefitted other species, by extraodinary transfers of genetic material that have become included into our or their genome. We have "Stolen" uselful stuff from yeast and snakes, for gawd's sake. And these kinds of events are the originally claimed source of the novel virus we call Covid-19.
I support the idea of choice in medicine generally. Informed choice gives us a chance to be responsible for ourselves. Political mudslinging to create intolerance even in a public health setting is in most scenarios just a negative. Not all, because some things are just that dangerous or deadly.