No. None of the vaccines are made up of materials that can cause disease.
Both the Pfizer/BioNTech vaccine and the Moderna vaccine use a small, harmless part of the virus’s genetic material called mRNA. This is not the virus. mRNA is a spike protein, and mRNA vaccines teach our bodies to create virus proteins. Our immune systems develop antibodies against these proteins to help us fight the virus that causes COVID-19, if we become exposed to it. This called an immune response.
The Johnson & Johnson (J&J / Janssen) vaccine is a viral vector vaccine. Rather than mRNA, it uses the harmless "adenovirus" as a vehicle to introduce the coronavirus's genetic material to your immune system. The adenovirus inserts the material into your cells to produce the distinctive coronavirus protein, much like the mRNA vaccines. This again creates the coronavirus protein to be shown to your immune system, triggering the immune response that teaches your body to create antibodies to help us fight the virus that causes COVID-19, if we become exposed to it.
As with any vaccine, you may experience some symptoms after receiving the shot. This is also an immune response, and is how we know the vaccine is working. The most common after-vaccination symptoms include a sore arm where you received the shot, headache, chills, fever, or fatigue. These symptoms shouldn't last for more than a day or two, and over-the-counter pain relievers / fever reducers may help.