I would be difficult to replace the internal CRT with an LCD. But it would be relatively easy to use an external VGA display, even an LCD display if it has the standard 4:3 aspect ratio (not widescreen) with VGA input. You can do this on the older tray-loaders, but not the newer slot-loaders.
On the tray-loader iMac G3, if you open the bottom panel, you will see two or three connections (the third one is for the IR sensor - not present on all tray-loaders). One of those connectors is the internal video connector. It is the one that looks like an old-style video connector with the two thumb screws. You can test to confirm the problem is on the analog video board by disconnecting that internal video cable and attempting to start up your iMac. You won't see anything in the display, but if it should sound like it starting up. Once it sounds like it got to the Desktop, you can press the power button and then Return to shut down.
However, that internal video port is not a VGA connector; it is an old-style Apple video port.
You need to get one of these adapters
[Apple Mac to VGA switchable video adapter TVCS-183|http://cgi.ebay.com/Apple-Mac-to-VGA-switchable-video-adapter-TVCS-183
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There are other similar ones, but I know this type works. Connect this type of adapter to the internal video port on the iMac. Then, you can connect a standard VGA cable and display to the adapter. The iMac should start up now, because you are bypassing the hardware failure on the analog video board. With the 333 MHz model, it has a whopping 6mb of VRAM and should be able to go as high as 1600x1200 at thousands of colors, or 1280x1024 at millions of colors (if the VGA display is large enough to handle it). The standard 1024x768 (and lower) will also work, of course.