"I was going with сигналить but my dictionary gave me "signal" for that which I interpreted as in signal left or right."

Left or right lamp int the car is 'поворотник'. One can say 'посигналил поворотником' or 'поморгал поворотником'.

'I think I've seen бибика - which I guess may mean 'horn' - in children speak?'

No, it's for car itself.