Learn Italian: Il doganiere cerca sempre il contrabbando. - The customs officer always looks for contraband.

In Italian, cercare is a transitive verb, meaning it takes a direct object without needing a preposition. The "for" in "looks for" is already built into the verb itself, so it's simply cerca il contrabbando.

Could sempre go at the very end of the sentence here?

It is much more natural to place adverbs of frequency like sempre (always) immediately after the conjugated verb (cerca sempre). Putting it at the end of the sentence isn't strictly forbidden, but it sounds clunky to a native ear.

Does contrabbando refer to the illegal goods, or the act of smuggling itself?

It can mean both! Here it translates directly to "contraband" (the illegal goods the officer is looking for), but il contrabbando is also the general term for the crime or act of smuggling.

How would the grammar change if the officer was "trying" to do something instead of "looking for" something?

When cercare means "to try" or "to attempt" an action, it must be followed by the preposition di plus an infinitive verb. For example, Cerca di trovare il contrabbando means "He tries to find the contraband."