"vote" vs. "vote on"

Solution 1:

"Vote" does not usually take a direct object, except in the comparatively rare case where the object is a word or phrase which your vote may be taken to express; so you can "vote 'Yes'" or "vote 'Reopen nominations'".

Normally it takes a indirect object of either of two types:

• vote for an option ("I voted for Jones");
• vote on a question ("I voted on the motion to close");

You can also vote in a poll or election, but I would not count that as an indirect object, but as an adjunct.

There is a more complicated construction which may be analysed in different ways: "They voted him into office". This might regarded as taking "him" as a direct object; but it is impossible without the complement "into office".