yes.
the Arduino can control a 'driver' that powers the motor. the driver is a device that takes high power and feeds it to the motor. the Arduino does not have power outputs, only logic I/O. Common application for an Arduino.
the spray heads might be solenoids. they can be powered and controlled with relays. like a driver, the relay takes the high power but is controlled by a low power logic signals. Also a common application for an Arduino.
I do not understand how you drive in a straight line, but an encoder can be used to show rotation, with the diameter of the wheel being known, each angle of rotation would equal some distancea simple ceiling mounted IR beam/break should work.. the encoder is also well suited to be used with an Arduino.
the pictures, as you said, are to be done with a different device. a laptop or some such.
the photo evaluation is beyond the abilities of the basic Arduinos currently used.
the Raspbury Pi has a lot of development in that area.
it would seem the only need is to have the Arduino tell the cart to move based on the lap-top and then to control the solenoids, again, based on signals from the laptop.
seems like your project is workable as you laid it out.