Is there an online database of acid and basic dissociation constants, all in one place, free to use?
There are a number of resources available that provide dissociation constants for acids and bases. The following are examples:
UC Davis Chemwiki pages E1: Acid Dissociation Constants at 25°C and E2. Base Dissociation Constants at 25°C provide many examples. These are based on information taken from CRC Handbook of Chemistry and Physics, 85th Edition.
The WikiBook Chemical Principles/Solution Equilibria: Acids and Bases provides a reasonable database of dissociation constants of acids and bases (Tables 5.3 and 5.4), as well as some further information about how the values are calculated.
The California State University webpage p-functions for common acids and bases, also at 25C, of a series of acids and bases.