From what I can tell, the vocals are live. I think the music however is from the studio recording. The rest of the band is pantomiming. Things like this were common in the 60s and remain so today.
Next time you watch a singer on Saturday Night Live, ask yourself if the band is really in the mix. Chances are that they are not. Sometimes the performance looks and sounds truly live, but the band is just pantomiming to a recording they made on the same exact stage the day before. I’ll save the ethics conversation for later.
I am fairly certain that Grace Slick is really singing live to the audience here. In fact her performance is sublime.
Happy Easter! Follow the White Rabbit!