Filemaker Pro 10 will work on Snow Leopard, at least as far as basic facilities are concerned - I can't answer for any web-based facilities. It's way the best database available, but likely to be expensive, even if you can find a copy on third-party dealers. It would have the important advantage that the documents could be updated to a later version of Filemaker should you move to a more recent computer in the future. Filemaker also runs on Windows.
iData3 and its update iData Pro 4 is a basic but reasonably flexible database: v3 will run on 10.5 up, v4 requires 10.7 up; it's $69.95 and there is a version for iOS. There is a simplified 'Lite' version for $29.95. Both can be used in a trial version for 30 days of actual use.
LibreOffice (free) has a database module but I've found it awkward and difficult to use: some people like it; and OpenOffice.org and its close relation NeoOffice both have much the same database module
If you can install Rosetta from the original install disks you could run AppleWorks on it, though you will have difficulty funding a copy now as it was end-of-lifed years ago. It has an excellent database module, surprisingly powerful, and similar to (though less powerful than) Filemaker. However there are disadvantages to locking yourself into such outdated software.
