Disclaimers: I am not a professional appliance repair person. I did sell appliance parts over the phone 20+ years ago. I don't have any connection to any sites mentioned.
To add on a little - sometimes there's something you can replace yourself, or have your handyman / maintenance person replace, without needing a full service call from an appliance tech. To do that, you have to come up with the replacement part.
Getting parts:
There are Web sites that are pretty good at helping you find the part you want; a couple of them are Repair Clinic and Parts Select. Even if you don't buy the part from them, that tells you about what the part costs when it's on the other side of a UPS truck.
The big-box stores like Home Depot or Lowe's will stock a few really popular parts, like electric range elements, knobs, water filters, etc. Some of the knobs come with multiple inserts and scales, so they will work on different brands and models. They might not match the other knobs or the appliance, but you can buy them at 8 PM on Sunday and get your tenant back up and running.
Most big cities will have a store that just sells appliance parts. So far, I have found that when you find the right store, their price will be pretty close to the online price. You pay a little more for the building and being able to walk out with the part today, but often that's worth a few extra bucks. Sometimes the price is different if they think you're a homeowner, vs a landlord, vs a service tech. Walking in there with your model and serial number (or part number if you have it), ready to go, helps. Spending 20 minutes asking the person at the counter to diagnose it for you hurts. Showing up in your work clothes helps. :D
Appliance stores will also have some stock of the most popular parts - more than Home Depot or Lowe's has on the shelf, but not as many as a dedicated parts-only store. Usually it works better to go to a "Bob's Appliance" type of store for this. Their prices may be a little higher than a parts-only store, but again, you can have it in your hand today.
Some appliance manufacturers will sell some parts directly on their own websites - usually the stuff that's really easy to replace or install, like shelves for a fridge or oven, knobs for a range, water filters for a fridge, etc. These are usually at list price (high), but it might be the only option if your appliance is really new.
Ebay and Amazon are more of a mixed bag. I tried to order some filters for the range hood in my rental from a seller on Amazon; they never showed up, so I got a refund. A few weeks later, they showed up in the mail, but when I tried to install one, it was the wrong size. I have since been ordering them through a local store; they cost more than Amazon, but they are genuine parts that fit right.
I have had a little better luck on Ebay; I bought a new motor for my washing machine at home, and it turned out to be a genuine part at a good price. Standard Ebay rules apply; just because the appliance manufacturer's part number is in the ad, and the photo of the item shows it sitting on the appliance manufacturer's parts box, doesn't mean it's actually a manufacturer part.
A little more on fridges:
The fridge in my rental is the first one I've owned with an electronic control board in it. It can do some tricks if you open the door and operate the door light switch a few times in a row, like turning the various motors and the defrost heater on and off in sequence. You have to be comfortable swapping parts to do anything with this information, but it might help you decide what's going on without taking the fridge apart. The sheet of paper that's folded up in a little envelope, behind the grille at the bottom front, tells you how to use the test modes. Other appliances with electronic controls may have similar test modes.
Because of that electronic board in the fridge, I bought a small one-outlet surge protector, and plugged the fridge in through that. It's been OK for about 3 years now, but ask me in 5 or 10 years if I think it helped. :)