Firefox will not launch the game for you, only Internet Explorer will launch the game if its installed properly. Could you take a screenshot of the problem and post it please? If you mean the actual start button on the site won't appear or summon the game client for you, that means you haven't installed everything that is needed or it installed improperly. Are you absolutely sure you installed Active X and D-Player?
First you download Active X. I would advice doing this in Firefox. Once downloaded and installed, turn off Firefox and if you have it open, IE as well. Then reopen IE on the digimon site and let it sit on the page AFTER YOU LOG IN until a pop up comes up or a little bar asking you if you want to block the site/add-on or install. If you notice the popup appears before you are logged in, simply X off the tab of Digimon and completely reload the page -- do not turn off Internet Explorer, just the tab of the site -- and then wait for the popup to appear either on the top (should be that way for you) or the button of the page.
Click install/allow popup (etc etc) and let D-Player install. After D-Player installs (make sure its installed in the same location as the game and stuff), press the START button on the front page and then you will get the actual GAME CLIENT install panel that will look like this:
If you are sure you downloaded everything and installed properly, it might not have went through anyways. That happens to me sometimes on Windows 7. To see if the programs installed, press the Windows START button to the bottom left of the screen and type into the search bar "Active X" or "D-Player". Or you can check recent programs in the same window.