Not sure about elsewhere but here in Scotland you can. My great-aunt died and it was less than a week later we got the okay to collect the medical certificate from the Sheriff Court. The reason I remember is we asked what to do next meaning could we go straight to the registrar. She must have misunderstood as she said 'Oh, I'm sorry you can't take her with you now.' Being a cheeky wee cow at the time and not knowing when to be quiet, I said that was a shame as we had the 20p bus fare all ready to bring her home. I'm not sure what was redder her face or my ear where my aunt whacked it.
Anyways, we went straight to the registrar who logged everything and according to her, as soon as it was in the system anyone could order it. It's the same with marriage/birth/divorce etc licenses. As long as you know the details, you can request a copy.