In light of their arrival on Shabbos, Rav Binyamin Chuta, a renowned posek, noted that the opinion of the Ksav Sofer is that a melacha intentionally done on shabbos for other people is prohibited to benefit from forever. This is a form of a knas, or fine, so that people will not intentionally violate the Shabbos rules. According to this opinion, Rav Chuta says, one should prefer to wait for the next batch of vaccinations to arrive and be in use before getting the child vaccinated. 

However, according to the Shulchan Aruch a violation of Shabbos for other people is permitted for benefit after Shabbos. Additionally, even according to the Ksav Sofer above it might be ok because the people who brought the vaccinations did it on the premise that it was permitted due to being a form of pikuach nefesh - if they thought it was allowed it is not considered an intentional violation of Shabbos but would be considered a shogeg, an error, and then even according to the Ksav Sofer the vaccinations would be permitted for use. 

Rav Chuta concluded that one who wants to be strict and wait for the next batch will be blessed, but one who wants to be lenient has what to rely on. Another factor to consider is that we don't know at this time when the next batch will be coming to Israel and it might be dangerous to wait for that. Anyone wondering what to do practically should ask a rav.

source: Hamechadesh