We have, but you're talking about interstellar (as in between the stars, or solar system to solar system) travel. The majority of the asteroids we have access to are locked up in the belt between Mars and Jupiter. We occasionally see comets, but they're not stable enough to sustain is, and then we have the forementioned need to get going at least as far as it is.

Something like ʻOumuamua, the "interstellar" object that moved through our solar system a few years back would be more on track with what you're thinking of. But that doesn't mean we could catch it. ʻOumuamua was 'tumbling' , as in it didn't have a fixed axis we could really locate to even think about trying to run up alongside it.