My father and I both use flight sim. Long story short I was wondering if it is possible to run Active Sky and REX off my server and both of our computers be able to run it?

Not at the same time, no.

 

With SimConnect your flightsim is the server and ASN is the client. One client program can only connect to one SimConnect server at a time.

 

You would have to run two ASN instances on your server, which could work if you use VMware or VirtualBox. But the ASN license check might prevent this.

 

One thing you could do is, to synch the weather between two simulators by connecting them over the network. Don't know if the standard sim multiplayer can do this, but there are addons that can. Like my FireFighterX for example, that is how I know. But I have to admit, I never tried this with complex ASN weather. Sounds interesting though, I need to check that.