If someone who isn't me wanted to open the little service panel on the top of a coin operated laundry machine similar to the pic, how would they determine the right key, and where would they buy it?
What would they find inside the panel, and how does it work?
It's a tubular lock and there notoriously easy to pick. You can either use some foil rolled into a tube and gently push it in repeatedly while turning to impression a key, or you can use a tubular lockpick available from basically any online retailer for ~ $25
Nice thing about the pick set is that you can lock it in place after it's decoded and either use it again, or use the decoded pick to cut your own key.
Note that the rubber band versions tend to work better than the "lock it in place" ones. (Source: I have both)
I'm assuming you own these machines and are forced to pick them open because you lost the key - if so, absolutely get yourself a tubular lock and a cheap bike lock to practice on before going to the coin op. Tubular locks are easier to pick than some padlocks, but they take practice to know how much force to use.
I'm assuming you own these machines and are forced to pick them open because you lost the key
That definitely would be my assumption. Otherwise it would be stealing from the land-lord, hoo boy, we wouldn't want that would we?
Hell of an assumption to make 
If you have the model number then you could search on eBay for the master key. Beware potential security cameras in the laundry area
Take a pic of the top of the tube lock on top of the washer there and tell me the model and I'll find you the key. You'll have to fiddle around inside of that maintenance access hatch but usually you can just figure out what the coin slot pushes against and just push that thing manually to start the washer.
look up a model number on the washer and buy the key for like 10 bucks.
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