Can I go to any auto parts store to get my vehicle scanned?

If you have a check engine light on and suspect it's something engine-related or with the ECU then any auto parts store should be able to scan it with no issue. They typically don't charge for a check engine light scan.
 
However, if you're having issues with the TCM, ABS, or any other computer module, you'll need to get access to a better scan tool to diagnose those vehicle computer modules.
 
You can purchase a very decent OBDII scan tool for around $100 give or take. Any auto shop with professional scan tools will usually charge roughly around that price range.
 
If you'd like to learn more about the differences between various OBDII scan tools, watch this video: