"Barbie Doll" was written in the seventies, and we know how "free" folks were looking to be back then. So it makes sense that Piercy would opt for free verse over a particular rhyme and meter for her poem. Plus, since she's talking about how culture often traps young girls into thinking and looking certain ways, it also makes sense that she'd look to exercise freedom in other ways, like meter.
Instead of fancy meters, we see a lot of enjambment, where one line runs on into the next. You get examples of this in lines 1-4, that lists all of the ways society tends to brainwash its young girls ("pee-pee dolls," "miniature stoves," "wee lipsticks"). Then we have other instances (lines 9-11) of quick end-stops between lines that keep the speaker sounding short, sweet, and to the point. The combo of both makes Piercy's poem sound more natural and honest rather than planned and prescribed, which allows the poem's message to ring a bit truer in our ears.
But we do have an instance of perfect rhyme in lines 2 and 4 ("pee-pee" and "candy") that reminds us of some of the more formal conventions of poetry. In fact, the overall sound of the poem has a nursery rhyme vibe to it, especially while reading the more macabre portions that include the girl cutting off her nose and legs (17-18). So although the poem sounds free, we do have instances of the speaker paying a sort of homage to the classic fairytale and perfect rhyme, while giving us a darker, more modern twist.
When we consider the context and message behind the poem, it also makes sense that the speaker would use free verse. "Barbie Doll" reads more like a story than a poem, so by keeping the form and meter free, we hear more of the telltale signs of a good story rather than a fancy poem. In other words, we're not distracted by too many singsong rhymes and perfectly timed syllabic patterns that might deter from the poem's message.