I was curious about that but never had a chance to test that. But recently the ejector spring broke on my Remington 870 and that was a good opportunity to test the shotgun without the ejector spring.
It will work! The shotgun was working properly but with much bigger chance of the stove pipe problem. I had a stove pipe for about each 5-10 shots. So yes, you can still use your shotgun with broken ejector spring in an emergency situation but you need to replace the ejector spring as soon as possible.