I'm looking for a cost of housing by zip code; even better if I can get data on 1-bedroom apartments.
For clarification: I would like to query over many thousands of zip codes, and not just one at a time.
HUD provides a dataset on average rental (50% percentile) on a county basis. I know that there is public data down to the zip code, but not sure where to find it yet.
There are a couple of options that you could look into. They both have trade-offs.
The first one is data from Zillow. I checked some of the data you're interested in, and it looks like it has a very small coverage of zip codes. On the plus side, if it has the zip codes you're looking for, then it would be timely.
The other one is from the Census Bureau. While there is a table that breaks down the price ranges people are paying for rent by bedroom for ZCTA, and the Median Gross Rent by ZCTA, it does not seem to have the precise estimate you're looking for. The downside with this data is that it is a pooled sample ranging from 2010 to 2014 which makes it less timely. The upside is that it is based on a nationally representative survey of nearly 11 million households. Another pseudo-downside is that the area is ZCTA and not zip codes, which are the Census Bureau's attempt to reconcile the fact that zip codes are terrible for research purposes.