For a while now I have been trying to reduce my intake of fried food, or food otherwise cooked with oil. However I still make a lot of things with onions, and I end up sautéeing them with vegetable oil. If I use a non-stick pan, can I sautée onions without oil? What are some techniques to cook onions without oil?
I would not recommend attempting to cook onions on the stove top without oil.
Another answer suggests that it's a bit like grilling; in theory, it is, but a traditional outdoor grill has a minimal contact surface. That minimizes sticking and makes it fairly difficult to burn food by contact alone. Using a dry fry pan for onions is more akin to searing, except that you generally only sear meat or fish. If you try with an onion, you are likely going to burn it before you ever get to the sweated/caramelized consistency you want.
You really have two options here. The first, and the one I highly recommend, is dry roasting in the oven. Cooking an onion involves the Maillard reaction and the dry, radiant heat of an oven is great for promoting it. Simply toss the onion on a baking sheet and roast it inside the skin for about 20 minutes at 450° F / 230° C. It will peel easily and you'll have a nice, soft, roasted onion. Don't overdo it or it will burn - the skin will help prevent this but keep an eye on it!
Your other option is to simmer or steam it, which will soften them substantially and cook some of the volatiles out (make it less pungent) but won't brown them because the Maillard reaction requires high temperatures and the boiling point of water is too low. Simmering should be obvious, just toss it in some water or stock. For steaming, either use a steamer basket or just put a small amount of liquid in a pan, heat it up to a simmer, place the onion in and cover it until cooked to your desired consistency.
Those are pretty much your only fat-free options. If you want to use a frying pan and expect to get them browned, then you'd better use at least a little bit of oil. If you're concerned about trans, saturated or polyunsaturated fats, use olive oil. If you're on an ultra-low-fat diet, use a blast of cooking spray. If you're avoiding "fried" foods out of an assumption that even a tiny amount of oil is unhealthy, I suggest consulting a dietician (if you haven't already) to validate that assumption first.
Aaronut's answer to a related question outlined the differences between non-oil methods of cooking onions (such as steaming or sweating). The taste will be different because different chemical reactions are occurring.
If you want to sauté onions without a lot of oil, try oil sprayers (Pam, or hand-pump sprayers loaded with your favourite oil). This will minimise the amount of oil you use. Using less oil means you have to move them around more often to prevent burning. (If you fry onions by themselves, they will probably just dehydrate and burn.)
Use a very small amount of stock to soften the onions (although as per Aaronut's linked answer above, you'll get a different taste).
Alternatively, if the rest of your recipe includes any meat, fry the onions when you brown the meat to take advantage of the oils which are already going to be present in your dish.
As an alternative to onions, try roasting some garlic cloves (or a whole bulb!) in it's skin and 20mins or so and use the resulting pulp in place of onion. It will lose a lot of harsh garlic flavour but still give plenty of onion quality (same/similar plant family?).
It might be worth drizzling a bit of oil when baking garlic to protect the skin, but I doubt too much would be absorbed.
A good quality nonstick saucier pan is a good choice to brown onions without oil. Start with a very fresh onion that has lots of it's own juice. Older dryer onions are not a good choice for this method.
Preheat the pan on medium. Let the onions sit untouched for a couple minutes on medium heat so that they may sear. Stir and let them sit again for a couple minutes. Repeat until onions are relatively dry and onion fond starts to build up on the pan. Deglaze with just a teaspoon of water to lift the fond off the pan just before it burns. At this point you cannot let the onions sit for long and you will have to stir often. Continue browning and deglazing until the onion reaches the desired consistency. If you want the onions dryer, you can reduce the heat to low and cover with a paper towel in between the pan and the cover to absorb the steam.
The pan should be small enough so that you cannot see the bottom of the pan. Otherwise the onions may burn rather than brown. I use a 2-1/2 qt Analon anodized copper saucier pan for one medium onion.
But I must say that just a spray of olive oil will make the onions darker with richer flavor and it will enable you to use stainless steel for better control.