the slack of his roundabout
There was a slight noise behind her and she turned just in time to seize a small boy by the slack of his roundabout and arrest his flight.
"There! I might 'a thought of that closet. What you been doing in there?"
"Nothing."

Question

Answer

A roundabout is a type of jacket, but the term is used nowadays. The child's jacket was loose in the back and she seized him by the collar.

See definition 3 here: http://dictionary.reference.com/browse/roundabout

Answer

Judy is right. A 'roundabout' is apparently a word used in the 19th century for a type of jacket. Here's some information about them:

http://www.customvestments.com/roundabouts.html

Answer

It sounds like a roundabout is an article of clothing that the boy was wearing. Some of it was loose, and that is what she grabbed to stop him.