I get this question--and others like "what age should my child start voice lessons?"--all the time. So, here's my answer.
It depends on what your expectations of the voice lessons are.
If you're hoping that your child will learn rigorous vocal technique and suddenly be able to sing "O mio babbino caro," then I'd say no. You should wait until the teen years to start with a voice teacher for technique. Just like every physical activity--dance, sports, etc.--there are age appropriate and inappropriate approaches. You shouldn't train a child to sing Puccini any more than you should train a child for a body building competition. It's physically unhealthy.
That said, I actually think singing lessons for children are a really good thing. Why? Because most kids are not getting enough exposure to live music. Children learn skills through exposure and imitation, but if they aren't listening to live singing how are they supposed to imitate? Listening to recordings is not the same thing. What children really need is musical immersion so that they can absorb musical concepts. Maybe your school has an excellent elementary music program and teacher, or maybe your church does. The more I move around the country, though, I'm convinced that this is clearly the exception rather than the rule. And even if your church or school have great children's music programs, could you say they're being immersed?
What should you expect out of a children's singing lesson then? Depending on the teacher/interest level, this may be an individual or group lesson. A good teacher will focus on teaching children music as if it's a language. They won't over-correct, just encourage experimentation with this new language. Kids can learn the language of music so much easier than adults. Most of the "tone deaf" adult students I work with now simply didn't spend much time singing as children and now they're working hard to learn something they could probably have absorbed as kids. Your child's singing teacher can teach concepts such as rhythm, placement, solfege, and melodic expression through vocal play. A child's music lesson should be fun, keeping him or her entertained through games, singing, stories, and other activities. Kids can learn to sing songs by themselves (solos) as well; again, a good teacher will pick age-appropriate repertoire such as folk songs or Broadway pieces intended for children.
Even if you decide not to go the formal voice lesson route, make singing with your kids a priority. Schedule a time and be the voice teacher; put together a plan and actively sing. There are plenty of resources on the internet for fun musical activities you can do with your children. Take the time to teach them a skill that will influence their mental development and stick with them for the rest of lives!