The funny thing is that injecting DNA is what viruses do, no nanobot are required though it does sound particularly devious if you use nanobots. Scientists have been using viruses to deliver DNA (and RNA) to genomes for years. Vaccines use weakened (non pathogenic) or bit of proteins to elicit an immune response so a using a vaccine doesn't make much sense. It would be far easier and far more effective just to blow around live virus particles in the air but showing people getting injected with life saving vaccine is much scarier.
Nature just ran an article today on the double dipping of vaccinations. Not only does a pneumococcal vaccination decrease the prevalence of the pneumococcus in Africa, it also decreases antibacterial resistance. This will actually save lives.
If you read the anti-vaxers posts, they urge you not to get vaccinations. This will result in more deaths of children. Ignorance is deadly and evil.