“The vaccines are based on untested, dangerous technology.”

This claim is centered mainly around the covid vaccines developed by Pfizer/BioNTech and Moderna, which both are based on mRNA technology. These are the first mRNA vaccines authorized or approved for use in humans, but the underlying research needed to develop them has been ongoing for decades. Scientists were testing out mRNA vaccines in human trials even before the pandemic arrived in late 2019.
The pandemic did accelerate the course of typical vaccine development, which usually takes years, but much of this speed was gained by the legwork already done on mRNA vaccines, the huge marshaling of resources, including financial, needed to conduct large clinical trials, and the expedited review of the data by health agencies—not by cutting corners in the trials themselves.