The problem is that that except for the very closest and largest stars, they are still just a point of light. A single pixel. So while you might get an image of the 3 stars, they would still be unresolvable points of light. Remember, Alpha Centauri (the largest) is just a borderline dwarf star like our own sun. The other two in the system are smaller, and much smaller. The next 3 closest stars are dim red dwarves, and again would be pixel sized. After that comes the Sirius pair, which are very bright and about twice as far away.
Something like the Andromeda Galaxy is scores of times larger than the full moon, so the light gathering ability, exposure time, and resolution of the Hubble can gain a lot. I'm not sure that is true about tiny stars like the Centauri triplet. Remember, the Hubble is not a really big mirror, in fact by earth standards it's rather puny. So especially with nearby stars, earth based telescopes with much larger apertures will get you more resolution (in other words, being able to resolve a disk, at least a little bit). What hubble has going for it is the clearest and most undisturbed imaginable "sky", and the ability to take long exposures with no need for adaptive optics.
I am not sure if Hubble has ever imaged either system (a Cen or Sirius), I'll have to search a bit.
But it's not really what it's best at.
BTW, there was a fascinating article in Nature or Science in the last month about a wide scale survey of the Andromeda galaxy (M31) and it's satellite M33 showing dozens of dwarf galaxies and their remains that it has eaten. I meant to start a thread on it. I'll have to dig it out and do that.