I'm looking for a website that shows the population sizes of a species (doesn't matter which) as a function of time at a geographic coordinate. Is there a government website or other free database with such information?
You can access the Imperial College global population dynamics database. They will have time series data at specific locations. http://www3.imperial.ac.uk/cpb/databases/gpdd There is a sister database as well that might be useful. http://lits.bio.ic.ac.uk:8080/litsproject/
These contain several hundred time series, and you can see a paper that used them here: http://onlinelibrary.wiley.com/doi/10.1111/j.1461-0248.2011.01702.x/abstract