Arduino does not work with sketch when not connected to USB

Please try to power your arduino from a 9V battery. Simply connect the terminals of the battery to the barrel jack (or cut an old wall wart if you have one and make an adapter). It's possible that your 9V wall wart source is too noisy. If the project runs using the battery, try looking for a different wall wart, or add a bunch of capacitors between 5v and ground. (One 100uf and a few smaller ceramics should do the trick)

I wrote some code on the Arduino and it works perfectly fine when powered by the USB. However, when I switch to 9V DC, the sketch does not run.

(1) The meaning of

" ... I do not have it connected to the Vin pin though, I have the 9V connected to the DC power supply outlet. ... "

is unclear.

If by "DC outlet" you mean DC INLET where you would usually connect a plugpack/wallwart ... supply, then that should work. Anything else may be fatal. Why not connect to Vin? Vin is the ideal and correct place to connect a 9V battery.

(2) You do not say what the 'sketch' does or uses or expects. If eg it expects or needs a USB connection then it will (of course) not work.

You say that the UNO "still runs" - but, what does that mean.

Run the examples / basics / blink program with USB and without.
Does the LED blink the same in both cases?
If so, the problem is more liable to relate to the USB connection than to the powering.

You can run other sketches to see if they operate with USB & 9V powering. Ones that do NOT access tx/rx are best for checking as they are less liable to interact with the USB port in some uncertain manner.

Report back.

Blink.ino

void setup()   
{  
    // initialize digital pin 13 as an output.  
    pinMode(13, OUTPUT);  
}  

void loop()   
{  
  digitalWrite(13, HIGH);   // turn the LED on (HIGH is the voltage level)  
  delay(1000);              // wait for a second  
  digitalWrite(13, LOW);    // turn the LED off by making the voltage LOW  
  delay(1000);              // wait for a second  
}  

In my project adding a Schottky diode too +5v line solved problem. You can use standard diode but has higher forward voltage drop