Learn French: Nous cherchons de l'huile au supermarché. - We are looking for some oil at the supermarket.

Huile starts with a silent 'h', which acts like a vowel here. To make it flow smoothly, French drops the 'a' in de la and replaces it with an apostrophe. You use this partitive article to mean you are looking for "some" oil, rather than the whole world's supply.

Could I just say Nous cherchons l'huile instead?

It changes the meaning! If you say Nous cherchons l'huile, you are looking for "the oil" — perhaps a specific bottle you lost or a brand you already discussed. Using de l' shows you are just looking for "some" oil in general to buy.

The English translation says "looking for", but I don't see a word for "for" in the French sentence. Where is it?

The French verb chercher (conjugated here as cherchons) translates to "to look for". The "for" is already built directly into the action, so you never need to add a translation of "for" (like pour) after it.

Why is it au supermarché and not à le supermarché?

Supermarché is a masculine noun, so "the supermarket" is le supermarché. In French, whenever the preposition à (at or to) meets the word le, they are forced to blend together into the single word au. You will never see à le written side-by-side.