COVID-19 - Is it okay to send emails this time for post-doc position?
Yes, by all means keep looking for a position. The future is unknown. Things may settle down or not, but it would be a mistake to wait until they do before you even begin a search. But email responses may be slower than usual and people may need to be more tentative. Expect that. 

I suspect things will be hard for a while, but no one expects it to be permanent. 

You can't win the game if you don't play.