How do I address a bedbug infestation?
Sounds like bed bugs to me.  Have you, or anyone else, noticed bite marks?  Sleep with a flashlight near the bed and if you wake up at night, use that to look for them (too much light will scare them away).

My suggestion is to strip the bed of all linens and wash them (you need heat to kill them, hot water and/or hot air from the dryer).  Remove and check the mattress and box springs and then check the entire bed frame.  Eliminate any paths from the bed frame to the wall or floor (placing the feet of the bed in a bowl of water or some other boundary).  Put the mattress and box springs in a zippered cover.  And finally, use a bed bug spray to kill any that make it through all this.  Note that these things can last a long time, so don't let down your guard for at least a year since you saw the last sign of them.