Honestly, I also often brought the kids to the pharmacy, lots of different pharmacies depending on what was convenient. Literally nobody *ever* questioned "Yup, we're here for the flu shot", whether they'd ever seen any of us or not except... actually, except for the mass vax event where E got her first covid shot. And even then, she was already in high school. I bet if I'd just said "Oh, yeah, I'm her mom, we just have different last names" and E had backed me up they would've gone along with it. But as her dad was right outside we just called him and he came in and gave his ID instead. (She wanted me to hold her hand while she got vaccinated instead of anybody else, because, again - all her shots, literally all of them, after infancy were with me.)

I'm not sure if other shots would've been different, it's possible they'd scrutinize more closely if it was tetanus at the pharmacy instead, but... in my experience, people just are not nearly as stringent about this as we'd perhaps like them to be.

Again, I'm not at all recommending this, and if LW did something like this they'd have to pay out of pocket to avoid insurance problems. Just as well, not having insurance is consistent with not having an ID.