Portable air conditioner exhaust through door
To quote from Consumer Reports, 


  Unlike a window air conditioner, all the mechanical parts of a
  portable air conditioner are sitting in the room you’re trying to
  cool. This contributes to the noise and less-than-capable cooling, as
  the portable unit is using conditioned air from the room to cool the
  condenser and exhausts the hot air out an ungainly exhaust hose that
  resembles a dryer vent. That creates negative pressure causing
  unconditioned warm air from surrounding rooms or outdoors to be drawn
  into the room you’re trying keep cool


So unless the rest of your house is cool, in which case a box fan will do the job, you're not going to get much cooling in your laundry room.   

In sum: yes, you can install a portable unit, and it will operate. No, it won't do much good. 

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