When sautéing vegetables, should I wait for the oil to heat up first?
In addition to the fact that some oils get quite a bit hotter than other, (so that hot with one oil isn't the same as hot with another), there is an in-between here that some cooks rank as a default option whenever inexperience prevails, namely warm oil. Regardless of which oil one is using, this can be achieved by applying an initial lower temperature to the sauté pan and then cranking up the heat once the vegetable(s) are in the pan and can be visually assessed for how they're performing.

Hot oil with vegetables high in water content risks splattering or worse (fire) when suddenly introduced. It is also easy to burn more delicate ingredients such as garlic (burn, in this case, meaning no more than browning). Meanwhile, ingredients porous in nature (such as mushrooms) added to cool oil are certain to absorb (and retain) more of it than you would perhaps hope for. Stated here, (The Reluctant Gourmet),


  There really is no ideal temperature. It really depends on what you are sautéing and what you plan to do with the ingredients when you are done.


Since it's not therefore difficult to see that a rule of thumb toward how hot cannot be established, (though no saucier recommends cool, nor the wiki on Sautéing), decisions must be based on knowledge which is specific to the ingredient(s). If uncertain, go with in-between.