A human cell line begins with tissue from a human person. The cells are grown in a container so that they replicate; they produce more cells of the same type. Then those new cells are moved to multiple new containers, so that more cells are grown, generation after generation. Quickly, it reaches a point where there are no cells taken from the source tissue, no cells that were ever in that human person. These cell lines can continue for many years, even many decades, so that the connection to that human person is quite distant.
Now, this can be done from cells taken from a living adult; a fetus is not required. It could be done from a fetus or infant who died of natural causes. But since many in the research and medical community believe that abortion is moral, they sometimes have used cell lines that began with an aborted fetus. This is much less common in recent decades, though one case of a cell line from an abortion is cited by the NCB Center (link above) from 2015.
Now for the ethics of the situation. First, it is important to know that the abortion did not occur in order to derive the cells. The abortion would have occurred in any case. Also, the use of cells from fetuses occurs in only a handful of cases over the past 50 years or so. The use of these cell lines in no way increases the number of abortions, nor encourages abortion. And once the cell line is established and is in use in research and production of vaccines, a decision to use the vaccine in no way affects that past abortion. Opposition to this source of cell lines is good, as it encourages researchers, health care professionals, and medical companies to consider the ethics of their decisions.
However, the decision by an individual patient to accept a vaccine is not, for vaccines against disease, intrinsically evil. Even if another vaccine is available that has no such morally-remote connection to abortion, it would not be intrinsically evil to use that vaccine (though it might still be immoral depending on the circumstances). When the Church teaches that an act (deciding to accept a vaccine from an aborted fetal cell line) is morally acceptable for proportionate reasons [Dignitas Personae 35], it is necessarily implied that the act is not intrinsically evil. For intrinsically evil acts do not become moral for a proportionate reason, not even for a grave reason. This means that the second font of morality, the moral object, is good: the use of a vaccine to prevent disease is the good object. And there is not second moral object which would be evil. The second font is good.
Can a vaccine be intrinsically evil? In the case of a vaccine that causes abortion, yes. But that is not what we are discussing here. This article is on vaccines that prevent disease, or cure a disease (in the case of a “therapeutic vaccine”).
Three good fonts are needed for a moral act. The person must have only good in their intentions. And the good consequences must morally outweigh the bad consequences. The bad consequence is that the rejection of such vaccines has (for each individual decision) a very limited effect on the medical community, to discourage this behavior of using aborted fetal tissue in the future. The good consequence is that harm from an illness, and in the case of Covid-19 grave harm or death, is avoided or greatly lessened, depending on the efficacy of the vaccine. This weighing of the consequences is usually left to the individual in each case, unless the Church decides to instruct the faithful not to accept a particular vaccine.
Now, in my opinion, a Covid-19 vaccine — if it were developed from such a cell line — would be entirely moral to accept, even if a different vaccine were available that was not from such a cell line. The reason is that Covid-19 has a substantial chance, not only of your death if you become ill, of someone else’s death. You cannot know if you are infected and spreading the disease until you have symptoms. Symptoms result on average 5 days after infection, and up to 14 days (99th percentile).
And the risk is not only to yourself. The average number of persons who become infected from each person who is infectious is estimated, on the low end, at 2 to 3 new infections. But one study put the range at “5.5 to 25.4” new infections from each infected individual .
You don’t have a right to risk the lives of others. That is not you being a holy martyr by turning away a vaccine that has a remote material connection to an abortion that would have happened in any case. (It would be quite a different moral situation if each choice to accept a vaccine resulted in a new abortion, or if each set of so many vaccines required a new abortion. But that is not the case; no, far from it.)
The reason that you could morally accept a Covid-19 vaccine from an aborted fetus cell line, even if another vaccine type were available, is that a vast number of persons must be vaccinated to quell the pandemic, and the production facilities for vaccines is very limited, by comparison. If many persons reject one vaccine, insisting on another, it may cause delays and disruptions, resulting in more infections and more deaths than if everyone simply accepts whatever vaccine is available to them.
Choosing between two vaccines: If one vaccine is from an aborted fetus cell line, and the other is not, choose whichever one is safest and most effective. Those factors morally outweigh the remote material connection to an abortion in the past, perhaps distant past, which would have occurred in any case. Choosing a less effective vaccine risks other persons’ lives unnecessarily. Choosing a less safe vaccine (such as live attenuated) risks your health, again, without any moral necessity.
Ronald L Conte Jr.
 Lauer, Stephen A., et al. “The incubation period of coronavirus disease 2019 (COVID-19) from publicly reported confirmed cases: estimation and application.” Annals of internal medicine (2020).
 Msall, Michael E., et al. “Severity of neonatal retinopathy of prematurity is predictive of neurodevelopmental functional outcome at age 5.5 years.” Pediatrics 106.5 (2000): 998-1005.