If Shabbat is the earliest an immunocompromised congregant can receive their first dose, should I encourage them to sign up, or keep looking for an appointment not on Shabbat? They don’t drive and can arrange transportation beforehand.
In the case of someone immunocompromised, and assuming that they don’t have to sign anything, they can arrange for it on Shabbat, provided that they make sure that there are no דאורייתא concerns. Specifically, the internal light of the cab has to be taken care of and payment needs to be arranged before hand. They also shouldn’t carry anything if possible. If they need their ID or anything else then they can have it on them.
If there is a heter for ploni to go on Shabbat, can this person’s partner (who is beginning [not official with the Vaad yet] the conversion process) go with them? Or, davka SHOULD the partner accompany this person to help emotionally support them and possibly help with any other melakhah issues that could come up?
Definitely go. Not even a question unless he is at a state of his conversion where he is doing all the mitzvot. Even in the case of a Jew, he could go along to provide emotional support if needed. Others can do issurei derabanan for the choleh as well.