Without a canner you are limited to canning high-acid foods.
Botulism spores don't die at 212F, the boiling point of water. A pressure canner boiling water at 15PSI raises the boiling point to 250F or so which will kill the spores.
The bacterium cannot grow in a high acid environment and so high-acid foods such as fruit and pickles do not need to be processed in a pressure canner. Look for recipes for such foods. As use2199 said they will involve boiling the jars for a while to kill things.
An excellent resource is the Ball Blue Book that can often be found near the canning supplies in grocery stores. It always calls for Ball products of course but it has a ton of good canning recipes and instructions.
Don't experiment. Botulism is not a fun thing. Your lips get tingly and then you die shortly aftwards.
There is no rule of thumb encompassing all salad dressings.
One "rule of thumb" which sometimes gets used is to look at the shelf life of the most perishable component. Frequently it works; sometimes, it is dangerously misleading. A mixture of the things you listed can have a longer or shorter shelf life of that of the most perishable component. An example of lengthened shelf life is mayo; it stays good for longer time than a cracked fresh egg, because the yolk gets pasteurized and the pH is lower. But there can be examples of the other thing happening. The classic is the homemade garlic oil: you can keep pure garlic and pure oil for months in the pantry, but once you combine them, you get a botulism risk.
I don't usually keep salad dressings in the fridge, but if I did, I would look at why the most problematic component has the shelf life it has, and decide whether mixing it will change the condition. For example, if I had a mixture of oil, vinegar and pure emulsifier: Oil keeps for months because it has no carbs, so nothing for bacteria to eat. Vinegar can have a few carbs, but not a lot, and it also has a very low pH, so bacteria die in it. The combination still won't have enough carbs for bacteria, and will still have a low pH. So the mixture will keep for very long time, just like pure vinegar or oil.
On the other hand, imagine mixing vinegar with honey and water. Honey doesn't go bad by itself, because the carbs are too concentrated for bacteria. The vinegar and water will dilute them, so this mechanism of bacteria prevention vanishes. The pH of the mixture will probably rise a lot too, because the vinegar gets diluted, so no protection on that front either. Thus, this mixture is apt to go bad much earlier than pure vinegar or pure honey.
This method requires that you make a new decision for each new dressing you make, and that you acquire enough knowledge to be able to make such decisions. If you feel this is too much effort or too risky, you can either start adding conservants to your homemade dressings, or just start preparing a fresh dressing for each batch of salad.
Best Answer
It's fine. Vinaigrettes last pretty much indefinitely in the fridge.
If the oil was exposed to too much light or air it might have become a little rancid. Smelling and tasting it before using it on the salad is a good idea. I wouldn't expect anything like that after just a couple months.
Catija alluded to garlic above. Garlic in oil is dangerous because it has an elevated risk of developing botulism- however in a separated vinaigrette all the particulate matter is submerged in the vinegar portion. More than acidic enough to prevent any botulism.