Control toy car transmitter with Arduino Uno
First step: Check VCC level of this transmitter IC. If it is 5V then it is safe to connect power and ground of the transmitter board to power and ground of your Arduino. Just to power them together from one single power source.

Second step: most probably one terminal of each button goes to separate pins of the transmitter IC which pulls them high up to almost VCC level through some internal resistor (10-200 kOhms), while other button terminals are connected to the ground. You can check it using any voltmeter you have around. If voltage level on unpressed button is less then 0.5V lower then VCC - it is safe to connect those button terminals to your Arduino data pins and set those Arduino pins as usual digital outputs. Of course if first step was ok.