First note that despite what the UN Special Rapporteurs (for Palestine) have called for, [in March at least] Israel had refused responsibility for vaccinating Palestinians who are not Israeli citizens, invoking the Oslo accords:
The United Nations (UN) human rights body has released a statement saying it's Israel's responsibility to provide equitable access to Covid-19 vaccines for Palestinians in Gaza and the West Bank.
The body says differential access is "morally and legally" unacceptable under international law laid out in the Geneva Conventions on the regulation of occupied territories.
But Israeli health minister, Yuli Edelstein, told the BBC: "We can also look into the so-called Oslo agreements where it says loud and clear that the Palestinians have to take care of their own health."
The Oslo accords - which Israel signed with the Palestine Liberation Organisation - give the Palestinian Authority oversight of public health under the principles of self-determination.
So Israel's own rapid vaccination programme did not extend to non-Israeli Palestinians at this point in time, at least.
By mid-June 55% of Israelis had received both doses of the vaccine, while only 30% of Palestinians in the occupied West Bank and Gaza had received one dose. There's also a significant difference between Gaza and the West Bank. By June only 5% in Gaza had been vaccinated. By September, 37% of Palestinians in the West Bank had received one dose, but only 18% in Gaza. In the latter area, there was apparently significant vaccine skepticism; the Hamas de facto government had to introduce a vaccine mandate. And according to the NYT, Israel did vaccinate [at least some] non-citizen Palestinians who were working in Israel:
Last spring, Israel gave vaccines to more than 100,000 Palestinians who work in Israel but not to the millions of other Palestinians in the West Bank and Gaza.