In a carbureted car does the ingition key need to be in contact for the fuel pump to send gas when you press the gas pedal?

If the carb is full of gasoline you should be able to see the accelerator pump jets streaming gas down the throat of the carb when you open the throttle.

If the carb is not full of gas you need to crank the motor for a mechanical fuel pump to fill the carb.

If you have an electric fuel pump the key needs to be cycled on for 10 seconds and off. That needs to be repeated 6 times to pressurize the fuel lines and get gas to the carb.

Pressing on the gas pedal will send a spray of fuel from the accelerator pump into the carburetor. You don't need to have the ignition switch turned on.

Carburetor vs fuel injection has NOTHING to do with when the fuel pump has power.

For most vehicles, it the ignition is in the 'off' position, the fuel pump doesn't have power. SOME older vehicles have a mechanical fuel pump.

Some carb cars did have electric fuel pumps. Those did need key on to pump fuel. They developed very low fuel pressure and it was the float level valve that would let the fuel into the bowl when the level got low. With the key off pressing the gas will inject fuel whether the key is on or off. As long as fuel is in the carb bowl/s.  

i would get a mechanic to check it out

Nearly all American cars with carburetors used mechanical - not electric - fuel pumps so ignition switch doesn't matter.  "Pumping" the gas pedal did result in delivering some fuel from the carb's internal fuel bowls/accelerator pump.

Not typically as there was a direct linkage between the gas pedal and the carburetor.  Pumping the gas pedal was a common way for someone to flood the carburetor.