I've been working on DDD as a secondary, personally. He covers most of olimar's bad matchups. The only matchup he does worse than olimar in is the Meta Knight matchup. He does really well against marth, does well against peach, ***** luigi, and those are really the only matchups most people have problems with, with olimar. I, personally, also have a wolf problem, which is cured by DDD.
Marth also covers olimars matchups pretty well, while doing better than him in the MK matchup (although some see the oli mk matchup as only 40:60). R.O.B. has problems against peach, though he beats luigi pretty bad. Not as bad as DDD though, for obvious reasons. R.O.B. also has a HUGE meta knight problem, to the point of where you'd be better off using olimar, if R.O.B.'s only your secondary. And he doesn't really cover marth either, as far as I know. But I could be wrong. Also, GW isn't too big of a problem for olimar. There's indifference on what the matchup actually is, but it DEFINITELY wouldn't be worse than 40:60, and probably isn't even that.
As for going only Olimar in tournament, it's definitely possible. You're still going to have a Meta Knight problem, and if you happen to run into a really good peach, or boss, you're ****ed. But there are very few matchups that he has a LOT of problems with. But if you're better than you're the best one there, it's not like a matchup would influence an upset. And there's people like RichBrown and Dabuz that have gotten 1st in reasonably large tournaments. Other than that, there's a lot of 2-5 places (especially AndyG who kept placing second behind M2K lol), but not many 1st places. A secondary is definitely a good idea, but winning it on your own is possible, for sure.