As someone who looked over all mellow thinking "I just love having a fire in the kitchen, so warm and cozy." ((used to have a fireplace in my kitchen))... to realize "YIKES!!! My oven is on fire!!!" Szwooooooooooosch of fire extinguisher. Double check to make sure that it's the alkeseltzer type of fire extinguisher (sodium bicab), and then having to scrub my entire kitchen to get the alkeseltzer off of everything. Don't avoid cleaning it.
((I cook a lot, and often at high temps - 650, etc. I've noticed most grease tends to light up right about 500 degrees.))
The self clean option on your oven will help to an extent... but it will smoke and smell awful.
If you don't like & or are sensitive to the fumes... can you get or pay someone else to do it?
And yes... both vinegar or an alcohol (like vodka, everclear, etc) can also help clean it... but avoid the alcohol in this case. It's possible to get alcohol poisoning form the steam as well as to cause an explosion from the vapors. I use alcohol to deglaze a LOT of kitchen stuff... but most pans only need a couple shots. So I'll only get a little drunk from steam I accidentally inhale. An over would need much more.