Parts:
Arduino Uno
Motor- 4Pcs SG90 9g Micro Servos for RC Robot Helicopter Airplane Controls Car Boat (Amazon)
Sensor- MQ-5
I would like to connect both the motor and sensor to the arduino board, but have been given conflicting opinions, and I'm not sure how to check if the power from the Arduino would be sufficient for the motor. Would I need a driver (transistor or H-bridge motor driver) and an external motor power supply?
Not for servos. The driver (H-btidge and PID for positioning) is built into the servo.
To add to the yes from @UKHeliBob, the power supply needs to be able to supply the stall current for the servos. A servo will pull stall current, briefly, every time that it starts. If all for can be starting at once you need to be able to supply the stall current times 4. It is usually recommended to supply minimum of 1A for each SG90 9g Micro Servo.
Draw up a circuit (block diagram will suffice) of what you intend to do given the advice you have already received.
Mark the battery supply you intend to use as well as motor type/size/controllers etc.
Show any power lines and signal lines separately.
This can then be used for any further advice if required.