It could be done perhaps with some slight modification but it's high risk, because your 5V supply may not provide 5V always (e.g. when switched on) and there's no protection circuitry either if you bypass the main supply input. It's a $500 board, during a time when component shortages exist... if you need to ask this question, I think you should just wait until you have a 12V supply.
It's a grey area if you replace with a lower rated supply, since that would be trying things out that the manufacturer does not state they support. And each time you have an issue, you'll be unsure if it is related to the supply or not.
Go for a 12V 5A supply if you cannot find a 12V 4A supply. 12V 5A is pretty common, there are lots of them available
