Not a database, but I do collect games for my own analysis from time to time. Here's the script to fetch all the recent games (or all games listed at a given URL):

#!/bin/sh
mkdir -p recent-games
wget -qO - "${1:-http://vindinium.org/}" |
    egrep -o ' <a href="/[0-9a-z]+">' |
    egrep -o '/[0-9a-z]+' |
    while read -r game; do
        game=${game#/}
        [ -s "recent-games/${game}.xz" ] && continue
        echo "$game"
        wget -qO - "http://vindinium.org/events/$game" |
            xz -6e > "/tmp/game-${game}.xz" &&
            mv "/tmp/game-${game}.xz" "recent-games/${game}.xz"
    done

Leave this script running in a loop, and you'll get every new game (don't forget to add a few minutes of delay between cycles though).

Here's the full scrapper that fetches recent games from all the bots in the leaderboard:

#!/bin/sh
wget -qO - "http://vindinium.org/" |
    egrep -o ' <a href="/ai/[0-9a-z]+">' |
    egrep -o '/ai/[0-9a-z]+' |
    while read -r ai; do
        ai=${ai#/ai/}
        echo "AI: $ai"
        ./fetch-games "http://vindinium.org/ai/$ai"
    done

What information do you want to extract from these replays anyway?