There are two big questions to consider: 

1. How do you define 'local'?

• Smaller geographic areas, e.g. data for ZIP codes, block or census tracts: the smaller the geographic area, the harder it will be to find detailed data.
• Medium to larger geographic areas, e.g. counties, cities, states: it will be much easier to find data for larger areas.

2. What kind of data for local markets are you looking for?

• Social or economic indicator data:  These are easier to find, by the way. 
  (If you're not sure what these are, here are some brief dictionary definitions: social indicators, and economic indicators.)
• Lifestyle or behavior-related data: These can be harder, or more expensive, to find.

This page focuses on resources for the US, and looks primarily at geographic areas smaller than the states:  this means ZIP codes, cities (or towns), counties, metropolitan areas (sometimes called MSA's, DMA's, and more). 

The types of demographic information you can find on these local areas often depends on who is doing the data collection.  For example, the Census Bureau, one of the largest data collectors in the world, does not collect much about people's buying habits or religious attitudes; for that variable, you will have to rely on data from commercial market research companies or on non-profit organizations that do surveys.

With a couple of exceptions (ACCRA Cost of Living Index and Rand McNally Commercial Atlas ...), these are a bit older but still useful.  Some are no longer being published in print, and the online versions are either expensive, or beyond our scope.  Sigh ...

These usually only provide small snippets of data. (Unless you can pay, of course!) But they can be a source for information about buying habits, etc.

With the exception of FactFinder (which tries to provide everything!), the sites below have a focus on smaller geographic areas.  There are many more sites available ...

The following sources all have ZIP code level data.  Please be aware that some variables are not collected at the level of ZIP codes, and you may need to consider using a larger, or different, geographic division.