A/V Crash Course for Church
Did you watch the webcam feed? What did / didn't you like about the quality of that? For some applications, webcams do "just work", and that might be good enough. Installing a "professional" setup can get real expensive real fast.

For audio, if the PA system is of high enough quality, picking up the audio from where you pick up the video might be good enough. The webcam mic likely isn't great for this, but there's lots of mics out there designed for video cameras that will also plug into your laptop to improve that. Get a USB audio interface as needed (google audio interface as suggested, I've used the ATR2USB, works fine), there are more specialist devices like the Rode AI-1 available if you want to use an XLR input. Getting a feed direct from the audio closet will be highest quality, but based on the distance, you'd either need to run a long cable, or invest in a wireless transmitter / receiver. So it all depends on your current webcam quality, versus how much you want to spend to improve that.

For video, do you need more than one camera angle? Do you have existing video cameras available? Again, something that can get expensive quick, depending how far you want to go. You can interface via something like the ATEM Mini (haven't got one myself, and no one currently has stock!), some cameras do have direct streaming available as well. If it's a fixed shot and the camera doesn't move, the webcam might even be fine. Maybe post some more detail about what does/doesn't work with the current setup, to get some more specific suggestions.

Also try browsing this YouTube channel: https://www.youtube.com/user/makeradventures - he's basically doing exactly what you are, and possibly has some content of interest.