Any predictions on who might win the private space race when it comes to Blue origin vs Spacex?

Your question is a good one. It is, however, my comfortable area of expertise. So, I contacted

Mark Baker

President, Temecula Valley Astronomers
Solar System Ambassador, NASA / JPL
Docent, CalTech Palomar Observatory

This is his answer.

So, the question was which private firm will win the Space Race?? So as posed, IF the SLS cannot support a Lunar landing by 2024, I feel SpaceX is in the best position to step in and assume the driver's seat. But this is just the first leg of a long and involved relay race...

I lean more to a collaboration of several private and government entities for the general race. Many private groups are focusing on integral parts of the endeavor, not the whole “enchilada”...in addition to Blue Origin and SpaceX, there are such companies as ULA, Boeing, Lockheed Martin, and Virgin Galactic with their hats in the ring domestically. 
However, you can’t discount any of the Chinese private enterprises such as iSpace, Landspace, OneSpace, etc. They do some orbital work but have already gone around and to the Moon with several missions!!!
India is in the race as well with their internal collaboration, ISRO, which you never hear about because they are quietly successful, but may be the first to return men to the Moon...
Israel has a non-profit Corp, SpaceIL, that has become aggressive of late and has a Lander on its way to the Moon...

The entity to dominate Space will more than likely be the one to develop the next evolution in propulsion that doesn’t involve the Neanderthal mindset of burning stuff...SpaceX and Blue Origin are both leading in this arena and their concept of reusability gives them a head start on expanding into the rest of the Solar System.

Insider note...NASA technologically is developing a drive system that has a million mile an hour capability. But it requires an orbital presence both around the Earth and the Moon, which Gateway will provide in the next decade!!!

Bottom line...the winner of the Space Race will be humanity regardless of who or what gets us there!!! But is Space ready for humanity...??!!!

 Chuck again.

I have the utmost confidence in Mark. I've known him for years, and he did an outstanding job of blending fact and opinion. I'd take this answer to the bank.

Thanks for using Authors Bureau ECN.

All the best.