Well, you would need to get a Blackmagic Decklink Quad for you inputs.
Potentially will need to use some form of HDMI extender or SDI cameras.
With SDI cameras you will spend a bunch and unless you really have a pro team and setup I'm just not entirely sure its worth it. A nice $1000US HDMI camera will be just about as good, especially for streaming purposes, and if you are getting several then that is a huge savings even with adapters to extend the HDMI signal. Blackmagic sells HDMI to SDI adapters for about $500 a piece but at around $1500 combined for camera and adapter thats still around $2000 less than the cheapest SDI cameras you will find. Money best spent elsewhere if you want my opinion.

One thing to think about is if you ever want to put the live video feed on a projector screen in the sanctuary/room/whatever that running the video through vmix on a computer is probably going to introduce more delay than you will get going through a standalone video mixer. I'm not trying to knock vmix, I think its great, but that is a limitation you need to consider.

Also, that computer of yours, if its not a quad core with a standalone video card, I'm not sure i'd recommend it. The ram is fine and the hard drive is fine and the speed of the cpu is fine as long as its a quad. Vmix depends on gpu capabilities a fair amount which is why it can perform so well. It doesn't need an extreme top end graphics card but its good to have a decent one. Streaming takes a fair amount of CPU resources which is why I recommend a quad, its just good to have plenty of cores to throw at everything.

Well, per the documentation, vmix adds about 2 frames of delay all by itself. If you are running at 50/60i/p that is 1/30th or 1/25th of a second. Add in any delay that the ingest card adds, which may be a frame or 2 itself, and you could be looking at a pretty noticeable delay. If you are using a slower frame rate, say 25 or 30, it will be extremely noticeable. If that is something that is important to you, you may want to look at a standalone mixer such as a Blackmagic ATEM. My church is using 2 ATEM TV Studios, 1 for video mixing and one for handling the projectors, and then using vmix for video streaming and recording to hard drive/dvd. The delay introduced by running the video feed from Camera to video switching ATEM to projector control ATEM to Projector is very minimal and you have to watch for it to really notice it.

I have never actually seen or used a decklink quad but as far as i know, you just install it, install the drivers, and off it goes. you get 4 inputs that you can use in vmix. The cheaper alternative is as many decklink minis as inputs you need but you need a system with that many pci express slots and while I assume they should all play nice together, I don't know that they would for sure.

Cameras, if it were me, I would get Canon Vixia HF G20's, run the HDMI into Blackmagic HDMI to SDI converters, run that into your mixer. The G20's can be found for around $1000US (if you are in another country than I am not sure what is available and pricing and you would want to talk to someone who knows more about that region) and are very good cameras.

As for running vmix on your mac, if its an Intel based Mac I would just use Bootcamp and use Windows to run vmix. There isn't another option for vmix.