It is almost always better to add a schematic to your question. Backwards engineering from a picture can be confusing at time and lead to incorrect assumptions.
What you want to do appears to be possible. It looks like you are injecting 9 Volts positive at VIN. Examining the Arduino Uno R3 schematic here we see the VIN net connected to the input pin 8 and also to the input side of the 5 Volt regulator (NCP1117ST50T3G). This distributes the 5 Volts to the Atmel processor (appears to power the LED directly) and to the external switch which is assumed to be normally open.
The total current needed by external devices connected to the Arduino Uno R3 5 Volt regulator should be considered. The NCP1117ST50T3G data sheet here state the regulator can handle in excess of 1 Amp.
Assuming a
10Kohm pull up resistor connected to the external switch.
However, Alkaline 9 Volt battery only have about 550mAh while Alkaline AA 1.5 Volt batteries have about 2000mAh.. You would need 6 AA 1.5 Volt batteries in series to generate 9 Volts. But they should last about 4 times longer.