Yes and no.
Plant-derived oils and animal-derived fat were used for lubrication in the whole history of mechanics and, surprizingly, also deep into the petrol era.
Even today, canola oil (with or without additives or processing) is used in motorsports as a high-performance 2-stroke oil.
What can go wrong?
Plant-derived oils have tendency to set (get more viscous all the way to forming a glass-like solid substance). The underlying process is known as polymerization. This is particularilly bad for parts with tight size tolerances - the already set oil makes impossible for the fresh one to enter the gap that needs lubrication. There are better and worse oils in this regard and canola oil is rather good, sunflower oil is in the middle and linseed/flaxseed oil is especially bad. Some additives can slow down the process.
The tendency to set is not of great importance for a frequently-used mechanisms, because the oil is frequently replaced with fresh amounts.
On the other hand, something that is left unused for a good half of the year can be rendered pretty much unusable without a great deal of disassembly and cleaning.
One can prepare their chanisaw for the seasonal hiatus by using 1 or 2 tanks of petrol-based oil just before the end of the seasonal work.
That's what I do in the related case of using vegetable oils as a diesel fuel. Older diesels (in hot enough weather) run pretty well on waste frying oil and other food-grade oils that lost their food qualities in one way or another. You just need to remember not to leave a significant percent of vegetable oil in the tank or in the feeding line before the winter.