The most cogent argument for the sinfulness of receiving these shots is that they give grave scandal by leading others to believe you and the Church are okay with aborting children in order to further medical research. Using fake docuмentation would give scandal in the same exact way. And it doesnt matter if its "only a venial sin!" We should all rather die than commit the slightest venial sin, and should never make light of violating God's law.
This being the case, I have heard the argument that since Catholic Churchmen are in disagreement about this and the traditional moral manuals make no explicit mention of it, the idea of the shot being scandalous is merely an opinion amongst others which need not be held... but still, if one is convinced that the vaccine itself is scandalous to receive - so convinced that they refuse to take it - then it seems to me the same scandal would prevent them from using a forged card.
Now if your reason not to receive it is solely based on the imprudent risks involved, or the argument concerning cooperation in the SIn of murder... then it's another question.