The first few years of a child's life play a key role in their development.
or should it be
The first few years of children's lives play a key role in their development.
Either is fine, but I find the sentence overall awkward. Perhaps a better way to state it would be
The first few years of life are a critical time in a child's development.
Or something similar. I don't understand what role a period of time plays in development. Parental touch, baby talk, nutrition, etc. are things that play roles in development. As an added bonus, you avoid the use of singular they, which I wholeheartedly endorse but your reader might not.