Absolutely. There are many ways of getting energy from the environment, and Earth has settled mostly on photosynthesis as the best way to do it. Two of the other, plausible mechanisms are chemosynthesis and retinal, of the Purple Earth hypothesis.
For example, the overall reaction of chemosynthesis looks like this:
$H_2S + CO_2 => H_2O + sugar + S_{(s)}$
No oxygen production necessary!
Photosynthesis dominates on Earth because it's so easy to obtain the required materials- you take in sunlight and CO$_2$, along with some trace elements, and you've got sugar. If you want a world where that doesn't happen, then all you need to do is make those materials harder to obtain. In an atmosphere that has negligible levels of CO$_2$, photosynthesis wouldn't be a viable strategy and organisms would have to turn to another source of energy.
As P. Chapman points out, whatever you use is going to produce a waste product. However, oxygen is an especially violent waste product. Its most common form, triplet oxygen, is a diradical- that's what makes it especially reactive with pretty much everything. That's what was so especially catastrophic in the Great Oxygenation event- it was essentially the equivalent of dumping lighter fluid on every other organism out there. With something like chemosynthesis, teh solid sulfur produced is pretty much nonreactive and would easily be sequestered, allowing chemosynthetic and anaerobic organisms to live together happily ever after.
I'll humbly link to my answer here for some other ideas about mechanisms for energy. Although all of those possibilities produce oxygen, it'd actually be easier for the reaction if it didn't. I had to use some questionable species like peroxides to make O$_2$ at the end, and if you just leave off those last steps you'll avoid the oxygen production entirely.
TL;DR: It's quite possible to avoid the Great Oxygenation event. Just make photosynthesis not worth it, either by limiting the CO$_2$ in the atmosphere or by decreasing the solar input to a point where it isn't worth taking advantage of.