file:// URLs work when the server is already mounted since it can just use normal filesystem calls. However, to access a server-based resource when the server isn't already mounted you need to tell the OS what protocol to use.
In most cases, assuming this is based on a Windows share, just replace file:// with smb:// to tell the Mac to use the SMB (Windows file sharing) protocol:
smb://servername/sharename/images/123.jpg
You'll probably also find that the Mac web browser doesn't understand SMB URLs since SMB isn't a protocol that most web browsers support, so at best they'll hand the connection over to the Finder. At worst they'll give an error message.
Either way you'll need to change the web app - either to use SMB URLs if Firefox supports them, or to use file URLs if it doesn't.
As far as the mounting question is concerned you can configure the Macs to mount the server at startup, either via a hard mount, or using automount to mount the server automatically as soon as any application tries to access data on it. This way you can mount the server at a consistent path that will make editing the web app easier.
The other solution that would require any client changes is to change all those UNCs to regular http URLs and serve the images via a standard web browser. This will work for all users on all systems and doesn't require any client-level changes. There's no advantage that I'm aware of to using UNCs in a web application. Since you have to change the web app anyway, this seems like a simpler approach that will continue to work in the future and doesn't require any special configuation changes that might be forgotton on new machines added to the network.