Why does my car shake and sputter while sitting idle after driving it as if it's going to shut down?

Unfortunately there are many factors that can cause this, and nobody can simply give an answer to this question. It may be as simple as needing a tune-up, or it may have a vacuum leak. it could be a number of sensors, bad coil pack, injector etc... If it has a check engine light on you need to start there. Just remember that there is a reason you pay a diagnostic fee for driveability issues at the shop. Trouble codes in most cases are a starting point of diagnosis, not a final decision on a part that needs to be changed. The most common example in my shop has been people that go to autozone, get their codes pulled for free and then buy O2 sensors to fix the bank1 / bank2 lean codes. Then when the light comes back on 15 minutes after they cleared it they bring it in and we have to explain to them why they just wasted almost $600 in one case replacing parts that were working, instead of fixing the vacuum leak. If you want it done right the first time take it to a shop. Not just any shop, make sure they have a good driveability tech.