Where are you staying? If you are staying onsite, both the DLH and the GCH have laundry rooms. They are not large but I have had good luck at both hotels doing laundry. Paradise Pier doesn't have a laundry room that I know of. At the DLH, the laundry room is near the quiet pool and workout room. At the GCH its easiest to just ask how to get there, its not near the pool area at that one. The one at the DLH had five washers and six dryers. It was a dollar to wash and a dollar to dry. They had a machine that sold soap and fab softner as well as a change machine.
If you are staying at one of the motels on Harbor, most of them have small laundry rooms near the pool areas.
I also saw a laundromat on Katella, less than a couple of miles from the Disneyland area on my way to Albertsons. So there are probably others nearby to DL as well.