On a world that had no oxygen (and thus is very unhelpful to would be metalworkers who require fire), could the natives find a way to use magnets to generate electricity? If so, to what extent? Is there some other way to form magnets capable of producing meaningful electricity other than with fire-based smithing, or perhaps a mostly-believable biological or other method?
To paraphrase a famous statement from The Graduate,
I have one word for you, just one word... batteries...
Chemical reactions aren't the most cost-effective way to create electricity, but we use them all the time. We also deal with a special class of non-metal materials that conduct electricity. They're called semiconductors.
However, this does not mean you're going to have a civilization filled with the glowing warmth of electric light and the easy listening of Motown soul. Without a good conductor, the potential of electricity (haha, an EE joke there. Did you notice?) is quite low. I doubt you could arc-weld without metal conductors.
Without plentiful oxygen there's a whole lot of things you can't do, like transportation (combustion), wastewater treatment, rocketry of pretty much any kind, plastics, acids (and a whole heck of a lot of other chemicals), and (not to make too fine a point) water.
No oxygen kinda means no photosynthesis, which begs the question, what kind of plants do you have? It makes for aerobic water (aka, life in rivers and lakes), so without it you're not fishing. In fact, insofar as we understand life, no oxygen == no life.
So, while electricity may be your main interest, it isn't actually your biggest problem. No oxygen is a bad thing.
If the goal is simply to produce electricity, then yes. You produce electricity without metal all the time in the form of static discharge. Rubbing two pieces of fur together or a piece of amber and a piece of fur together produce fairly impressive amounts of static electricity. As a matter of fact, peoples in the classical period observed electrostatic effects by rubbing cats with amber rods. In the 1600's when a chemist named William Gilbert first gave a scientific name to the phenomena he coined the term "electricus" which is Latin for "like amber" or "from amber," based on the greek term "elektron" which simply meant amber.
However, I get the feeling you are more looking for some sort of civilization that has harnessed electricity for useful purposes. I am sorry but there is no way to do so without possessing a firm grasp of metallurgy and metalworking.