Do I need a pull-up/down resistor for button LED circuit?
The circuit you have made is only using Arduino as a 5V power source. You aren't using any GPIO pins to read the state of the button or to control the LED by an output pin. You've just made a plain-old circuit with a button and LED in series with a power source.

Often for a button-controlling-LED sketch, you will have a button connected between GND and an I/O pin defined as an INPUT_PULLUP, which means the Arduino will be able to read the state of the pin as HIGH (5V) or LOW (0V). The PULLUP part indicates that when the button is not completing the circuit between GND and the pin, a weak voltage is applied to get it up to 5V. In this way it's not "floating" at an unspecified voltage, but is being actively driven to 5V. When you press the button, the input pin is shorted to GND.

A pull-up resistor causes an otherwise un-driven pin to go to logic HIGH. A pull-down resistor causes an otherwise un-driving pin to go to logic LOW.

In the case of a pull-up resistor, reading a logic LOW means the button is pressed, which is kind of the reverse of your circuit operation, so in your sketch code, you would look for LOW to turn on the LED.