I agree you need an "all of the above".
I selected "bring them with me" because that's the default. I TOTALLY agree with the idea that this is PART of homeschooling. This is "socialization" in a broader-society sense.
When I was a kid, I was extremely, debilitatingly shy, and one of my worst anxieties was dealing with any clerk in a store... even the friendly cashiers at McDonald's. I couldn't talk to them... not just because I was afraid to talk, but because I was worried I was going to do something wrong. I didn't understand the social rules, I was scared there would be some rule that everyone else knew but I didn't.
I'm not trying to imply that if I'd have been homeschooled that would have been automatically fixed.
I think I may have some Asperger's tendencies. I'm just pointing out how important those skills are, and that I had a hard time learning them.
So my kids come with me for all errands by default.
Of course, sometimes I really just would rather go by myself. If I'm bra-shopping, for instance, or even any clothes, and I need to spend an hour just trying stuff on, rejecting things because nothing fits my bizarre frame, trying more things... the kids get BORED SILLY doing this so it's best for me to do it myself.
Things like that, I'll either do when DH is home, or my SIL (who also homeschoolers her kids) will watch mine (we trade back and forth as necessary), or I'll do it when they're in some activity or other. For instance, when DS has his dance class, usually DD stays home with DH and I'll go grocery shopping then pick him back up.
It really all depends on what the errand is, whether it can wait or needs to be done now, what mood I'm in... heh...
But definitely, whatever your preferences are, take errands into account in your homeschool planning. Maybe you'd prefer to have an entire 'errand day' where you try to do everything one day a week, and so you don't schedule "official" school stuff that day. Or maybe you do school in the mornings, errands in the afternoon, like another poster suggested. Whatever you prefer, just do realize that it will happen.
And you know, I don't think this is necessarily a homeschool-specific question. Because it's not asking "how do you do errands when you're supposed to be teaching all day". It's asking "how do you do errands when you have the kids with you all day instead of away at school." So the same question could be asked of parents with public schooled kids, in the summer. They still get their errands done. So... so can we, year-round.
I always like in the summer when I see kids out and about with their parents all the time. That's the way it SHOULD be IMO... kids are a PART of human society and community and should not be segregated out of it as much as they are, as though they're completely separate and foreign until they reach adulthood.