I am particularly interested in housing costs; specifically 1-bedroom apartment rentals.
For clarity, I mean a database that's free (as in beer) to query with an arbitrarily large number of zip codes.
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Living Wage Calculator by MIT provides cost of living data down to locality, which you could mesh into zip codes, but since zip codes are not unique to a locality, you'd have to give or take something. perhaps use census zcta's? I'm not 100% on that, that's just the first thing that comes to mind. I don't see any bulk download options, but they do use a consistent naming convention across URI's and the data so scraping is relatively painless.
This might be helpful for other people if not for OP.
The problem here is that US Census data is based on census tracts, not zip codes. So for the US here is HUD "crosswalk" data from www.Huduser.gov that converts zip codes to census tracts, and vice versa. This will get you part way there, and may be very useful for other people. It's also recent, as of Oct 2015.
The crosswalk files will help you use US Census data by zip code.
And here are Fair Market Rents for the US states and cities, which will get you part way there also. Actual XLSX file with county-level data is here. It has columns for state, county, and metro area name along with several types of averages of gross rent.
FMRs are used for some gov't program to pay for peoples' rent. This DOC file is an overview of the data.
Try search in http://www.city-data.com/ you have many information there.
It isn't free, but the following is an option: https://www.unitedstateszipcodes.org/zip-code-database/
The data comes from the Census Bureau American Community Survey. The ACS is free to download, but it is a pretty large data set that can be didfficult to work with.