Is there a module that allows me to read data from potentiometers/flex sensors simultaneously?

Hi. I want to make a project, but I need more pins for potentiometers/flex sensors. I want to use all of them at once. It would be great If the module had documentation so I could actually use it.

You can use an external A/D, they are available with up to 16 channels at a variety of resolutions and interfaces, your choice?

@gilshultz, but they don't work at once, do they? Are there ones where no multiplexer is involved?

Now it depends on what @di55 means by "at once"? At the same nanosecond, within a few milliseconds?

See my tutorial on Multi-tasking in Arduino which includes a loopTimer() so you can tell how close to the 'same time' things are happening.
Also note that AVR A/D gives a false reading for the first one after switching selectors/references so do two readings and discard the first.
Here is some code for multiple readings averaged out to reduce noise

#define SAMPLES 20 //Number of samples you want to take everytime you loop
// The Atmel datasheets for the micro-processors Arduino uses for Uno/Mega2560 advises
// the user that the first A/D reading after changing the reference voltage may be incorrect and should be ignored.
// 20 samples takes about 2mS to complete
int readAnalog(int Apin) {
  long analogValue = analogRead(Apin); // allow for sum
  // discard the first one
  analogValue = 0;
  for (int i = 0; i < SAMPLES; i++) {
    analogValue = analogValue + analogRead(Apin);
  }
  analogValue = analogValue / SAMPLES;
  return (int)analogValue;
}

At the same time. Just so it isn't noticeable.

How many analog pins do you need?
With an Arduino Nano you get 7. Even some of the newer ATtiny series MCUs give more than that. Otherwise, as suggested, external ADC chips. Ensure an Arduino library exists for your chosen chip unless you want to write the driver yourself.