A line-marking cart? For painting the lines on a playing field?
Picked up my brush, walked on the lawn
Line marking equipment uses a brush to paint the lines onto the grass.
with lively pace at crack of dawn.
Carts allow the painting to be done faster than manually painting the lines, and this work is often done early in the morning before a playing field may be used.
A smart young driver, headlamps on,
I haven't seen carts with headlamps but perhaps they exist? Or maybe this refers to the floodlights on a playing field?
some lucent colours I have drawn.
Normally playing field lines are drawn in lucent, or light reflective colours such as white or yellow.