No, the mRNA COVID-19 vaccines do not contain any aborted fetal cells. Fetal cell lines are not the same as fetal tissue. Fetal cell lines are cells that grow in a laboratory. They descend from cells taken from elective abortions in the 1970s and 1980s. Those individual cells from the 1970s and 1980s have since multiplied into many new cells over the past four or five decades, creating fetal cell lines. Current fetal cell lines are thousands of generations removed from the original fetal tissue.
For the Pfizer-BioNTech and Moderna vaccines, no fetal cell lines were used to produce or manufacture the vaccine, and they are not inside the injection you receive from your doctor/nurse. Fetal cells may have been used to test efficacy and/or proof of concept (see sources below).
Johnson & Johnson
The Johnson and Johnson vaccine did use fetal cell cultures, specifically PER.C6 (a retinal cell line that was isolated from a terminated fetus in 1985), in order to produce and manufacture the vaccine.