This is probably a silly question, but when can I start entering my 2020 futurity horse?  Do I have to wait until December 1st or did it move to November?

You need to look at the rules for the specific futurity you are wanting to enter. They will normally have the dates listed. 

Its now Nov 15th for most.  But not all have adopted that date.  

Probably depends a lot on where you plan on running.  I think the southern and midwest futurities probably adopted the earlier date. You need to check the rules of the futurities you plan on running at.