Does anyone know of a website or program that can recommend openings that best fit your personal playing style? I'm envisioning something like a series of chess problems where, based on your answers, the computer spits out openings that are best suited to your play.

I only ask because while I'm great at finding openings I think are fun, I'm not so great at finding openings I can actually play well. Thanks!

I would say, start looking at games of the better known players.  When you find one that catches your eye, get a list of games from that person and go over them.  You don't have to "copy" them, but if their style fits your game, take some ideas from them.  Eventually you'll be able to tailor it to your liking.

The first and easiest answer is time, although that's a poor reason. Another reason is that at this level, whether I win, lose, or draw has very little to do with the opening and everything to do with who makes more mistakes in the middle and endgames. I had a pretty good success rate with the Saragossa Opening (1.c3) but I don't think anyone would seriously advise me to stick with it if I wished to improve my game. Furthermore, the more openings I look at, the more hopelessly bungled I get, as my rote memorization skills are poor and I prefer to learn in-depths the purposes and aims of an opening, rather than just the book moves. Learning the why behind the Najdorf, the Ruy, the KIA, etc. is an arduous and confusing task for me. 

Of course there's merit in putting in the work, but I have a short memory span and often lose sight of what I've previously learned when I move onto something new. 

There's also openings with easy schemes to learn with whites as the London System for example.