There doesn't seem to be anything in OE that will minimize it to tray.
Mozilla has an inbuilt mail client that can check automatically when it is closed. You may want to try that.. it'll import your outlook express settings, too.
Alternatively, I use something called ActiveCaptions which adds a right click feature to each of the Minimise, Maximise and Close buttons. Right clicking min with send any application tray, Max will roll it up to just the title bar and Close will keep it on top of everything else. It's not free though and I have yet to find a free one that uses so little memory.