Learn Danish: Ved du hvad for et søm vi skal bruge? - Do you know what kind of nail we should use?

The phrase hvad for et acts like the common spoken version of hvilket (which or what kind of). It uses et because the noun it describes, søm (nail), is an et-word (et søm). If we were talking about an en-word like hammer, it would be hvad for en hammer.

Doesn't Danish usually swap the subject and verb in a question? Why is it vi skal and not skal vi?

The main question here is Ved du (Do you know), which does have the verb before the subject. The rest of the sentence (hvad for et søm vi skal bruge) is an indirect question acting as a subordinate clause. In Danish subordinate clauses, the subject (vi) always comes before the verb (skal).

Does skal bruge just mean shall use?

Literally yes, but in Danish at skulle bruge is an extremely common way to express to need or to have to use. So hvad vi skal bruge translates smoothly to what we need or what we should use in everyday English.

I remember the word søm meaning seam in clothing. Is this the same word?

They are spelled and pronounced the same, but they have different genders! et søm (an et-word) means a metal nail you hit with a hammer. en søm (an en-word) means a stitched seam in fabric. Because this sentence uses hvad for et, we know they are talking about the metal nail.