Any user can make their own answer Community Wiki, but only moderators can (directly) do this for questions. You have two options to make your question CW, the first of which is the more sensical route (I only post the second to make the answer complete, plus I find it amusing.)
As you said in comments above, you may indeed flag the post for moderator attention and ask that it be marked as CW.
To do it yourself, you can take advantage of the fact that any post, when it reaches more than 10 edits in the revision history, is automatically converted to community wiki. So in theory, to do this yourself you can wait until the 5 minute grace period on edits are up, then edit your question. Then wait another 5 minutes, then edit again.... And keep doing this until the 10-edit threshold is reached and the CW conversion happens automatically.
...obviously #2 is silly, but you asked if normal users can do this, and well... You can! But just go ahead and flag it for moderator attention; we'll mark it CW for you, no problem.
(As an aside, I'll note that I don't see any reason for you to have to mark such questions as CW if you don't want to. When a question is CW all its answers also become CW, which means you won't gain any rep for the question or the answer. Since you're putting effort into both, I don't see why you shouldn't get credit for it. But if you do want to make it CW, you're always welcome to ask and we'll do it for you!)