The Pfizer and Moderna vaccines do not contain any live virus. Instead, it uses messenger RNA to teach our cells how to fight off the virus. The RNA in the vaccines cannot change your DNA in any way. For more detail on how mRNA vaccines work, visit the CDC website.
The third vaccine, from Johnson & Johnson, works a little differently. From Yale Medicine:
“This is a carrier vaccine, which uses a different approach than the mRNA vaccines to instruct human cells to make the SARS CoV-2 spike protein. Scientists engineer a harmless adenovirus (a common virus that, when not inactivated, can cause colds, bronchitis, and other illnesses) as a shell to carry genetic code on the spike proteins to the cells (similar to a Trojan Horse). The shell and the code can’t make you sick, but once the code is inside the cells, the cells produce a spike protein to train the body’s immune system, which creates antibodies and memory cells to protect against an actual SARS-CoV-2 infection.”
There are no eggs, preservatives, latex, or pork products in any of the vaccines. They are halal. For a complete list of ingredients, see the fact sheets for the three vaccines that have been authorized by the FDA so far:
See this page from Yale Medicine for more details about the differences between the various vaccines. All three options have been tested extensively and determined to be safe and effective.