Using push-button switches without pullup/pulldown resistors?

Is it possible for me to use push-button switches without pullup/pulldown resistors? I may have made a mistake wiring push-button switches with pullup/pulldown resistors and the Pololu motor controllers don't work properly with switches. The motor controllers work without switches.

The motor controllers are shown in Pololu - Dual VNH3SP30 Motor Driver Carrier MD03A
The switches are wired like the schematic shown in http://arduino.cc/en/Tutorial/Button

#include <Servo.h> 
Servo servo01;
Servo servo02;
Servo servo03;
Servo servo04;
int InA1 = 22;
int InB1 = 23;
int PWM1 = 9;
int InA2 = 24;
int InB2 = 25;
int PWM2 = 10;
int InA3 = 26;
int InB3 = 27;
int PWM3 = 11;
int InA4 = 28;
int InB4 = 29;
int PWM4 = 12;
int switch01 = 38;
int switch02 = 39;
int switch03 = 40;
int switch04 = 41;
int switch05 = 42;
int switch06 = 43;
int switch07 = 44;
int switch08 = 45;
int switch09 = 46;
int switch10 = 47;
int switchstate01 = 0;
int switchstate02 = 0;
int switchstate03 = 0;
int switchstate04 = 0;
int switchstate05 = 0;
int switchstate06 = 0;
int switchstate07 = 0;
int switchstate08 = 0;
int switchstate09 = 0;
int switchstate10 = 0;
void setup() 
{
  pinMode(InA1,OUTPUT);
  pinMode(InB1,OUTPUT);
  pinMode(PWM1,OUTPUT);
  pinMode(InA2,OUTPUT);
  pinMode(InB2,OUTPUT);
  pinMode(PWM2,OUTPUT);
  pinMode(InA3,OUTPUT);
  pinMode(InB3,OUTPUT);
  pinMode(PWM3,OUTPUT);
  pinMode(InA4,OUTPUT);
  pinMode(InB4,OUTPUT);
  pinMode(PWM4,OUTPUT);
  pinMode(switch01,INPUT);
  pinMode(switch02,INPUT);
  pinMode(switch03,INPUT);
  pinMode(switch04,INPUT);
  pinMode(switch05,INPUT);
  pinMode(switch06,INPUT);
  pinMode(switch07,INPUT);
  pinMode(switch08,INPUT);
  pinMode(switch09,INPUT);
  pinMode(switch10,INPUT);
  servo01.attach(5);
  servo02.attach(6);
  servo03.attach(7);
  servo04.attach(8);
  servo01.writeMicroseconds(1500);
  servo02.writeMicroseconds(1500);
  servo03.writeMicroseconds(1500);
  servo04.writeMicroseconds(1500);
  delay(1000);
}
void loop()
{
  switchstate01 = digitalRead(switch01);
  switchstate02 = digitalRead(switch02);
  switchstate03 = digitalRead(switch03);
  switchstate04 = digitalRead(switch04);
  switchstate05 = digitalRead(switch05);
  switchstate06 = digitalRead(switch06);
  switchstate07 = digitalRead(switch07);
  switchstate08 = digitalRead(switch08);
  switchstate09 = digitalRead(switch09);
  switchstate10 = digitalRead(switch10);
  if (switchstate01 == HIGH)
  {
    digitalWrite(InA1,HIGH);
    digitalWrite(InB1,LOW);
    analogWrite(PWM1,255);
  }
  if (switchstate02 == HIGH)
  {
    digitalWrite(InA1,LOW);
    digitalWrite(InB1,HIGH);
    analogWrite(PWM1,255);
  }
  if (switchstate01 == LOW && switchstate02 == LOW)
  {
    analogWrite(PWM1,0);
  }
  if (switchstate03 == HIGH)
  {
    digitalWrite(InA2,HIGH);
    digitalWrite(InB2,LOW);
    analogWrite(PWM2,255);
  }
  if (switchstate04 == HIGH)
  {
    digitalWrite(InA2,LOW);
    digitalWrite(InB2,HIGH);
    analogWrite(PWM2,255);
  }
  if (switchstate03 == LOW && switchstate04 == LOW)
  {
    analogWrite(PWM2,0);
  }
  if (switchstate05 == HIGH)
  {
    digitalWrite(InA3,HIGH);
    digitalWrite(InB3,LOW);
    analogWrite(PWM3,255);
  }
  if (switchstate06 == HIGH)
  {
    digitalWrite(InA3,LOW);
    digitalWrite(InB3,HIGH);
    analogWrite(PWM3,255);
  }
  if (switchstate05 == LOW && switchstate06 == LOW)
  {
    analogWrite(PWM3,0);
  }
  if (switchstate07 == HIGH)
  {
    digitalWrite(InA4,HIGH);
    digitalWrite(InB4,LOW);
    analogWrite(PWM4,255);
  }
  if (switchstate08 == HIGH)
  {
    digitalWrite(InA4,LOW);
    digitalWrite(InB4,HIGH);
    analogWrite(PWM4,255);
  }
  if (switchstate07 == LOW && switchstate08 == LOW)
  {
    analogWrite(PWM4,0);
  }
  if (switchstate09 == HIGH) 
  {     
    servo01.writeMicroseconds(500);
    servo02.writeMicroseconds(2500);
    servo03.writeMicroseconds(500);
    servo04.writeMicroseconds(2500);
  } 
  if (switchstate10 == HIGH) 
  {     
    servo01.writeMicroseconds(2500);
    servo02.writeMicroseconds(500);
    servo03.writeMicroseconds(2500);
    servo04.writeMicroseconds(500);
  } 
  if (switchstate09 == LOW && switchstate10 == LOW)
  {
    servo01.writeMicroseconds(1500);
    servo02.writeMicroseconds(1500);
    servo03.writeMicroseconds(1500);
    servo04.writeMicroseconds(1500);
  }
}

Change these from

pinMode(switch01,INPUT);
to
pinMode(switch01,INPUT_PULLUP);

or
pinMode(switch01,INPUT);
digitalWrite(switch, HIGH);

either will enable the internal pullup resister.
If the switch is open, they will read as High.
If the switch is closed to Gnd, they willl read as Low.

Does Arduino Mega 2560 R3 board have pullup resistors on I/O pins? I made a test circuit on Arduino Uno R3 board and it worked. I think the circuit doesn't work on Arduino Mega 2560 R3 board. I connected the switch from GND to I/O pin 38 on Arduino Mega board and nothing happened after inserting serial.println lines.

Data sheet says they do:
"All port pins have individually selectable pull-up resistors with a supply-voltage invariant resistance."

Did you change your setup code to enable them?

I changed the code from

pinMode(switch01,INPUT);

to

pinMode(switch01,INPUT_PULLUP);

I think that I may have made wiring errors. I tested the circuit with two buttons and a servo on Arduino Uno board and it worked.

Cool.

I got the circuit working on Arduino Mega 2560 R3 board with 10 push-button switches and five servos.

Cool. Buttons with internals pullups only?

Buttons with internals pullups only? Yes.