You could use the same software you edit with, to capture a video input live, however, there are some possible “gotchas”.
1: are you shooting in HD or standard def, what is coming out of the switcher, is THAT output HD or standard def, and using what format and connectors? If your switcher output is HDMI, you’ll need an HDMI input on the recording device, or some kind of translator (transcoder) in between the two. AKA and Blackmagic and others offer those devices, and AJA advertises a recorder module you might like caled the KI Pro, but they are pricey.
2: Yes, you can capture into, for example, Final Cut from a live video source, you just tell the capture module you are capturing from an uncontrolled device, and you can manually start and stop a recording. But, chances are you could get a momentary glitch in the signal that will cause the computer to drop out of recording, particularly the longer the amount of time you let it run without stopping, and it takes time to re-establish the capture. There is software that Adobe bought and re-branded as part of Premiere as On Location, this would be a better way to capture to a laptop, and you could edit from there with premiere or another NLE. Liekwise, I think Sony Vegas might be used for this, on a laptop or on a PC, i believe it has a specific interface for live recording but I’m not an expert on Vegas or Premiere, so you’ll have to get someone else’s input on the relative merits there. With most of the computer based recording solutions you are going to need a substantial outboard set of AV drives in a RAID array to get the thruput and speed and capacity for a long program, as well as redundancy against loss of data. That’s an added expense and figuring where and how to connect the drives and line up a signal flow is an added concern. And do you REALLY want to take your editing system out to the location for this?
There are stand-alone DVD recorders and some of them have multiple types of inputs, but I can’t recall one that makes a “high def” DVD on the fly. I would say as part of your church stewardship, decide first if the added complication and expense of an all-HD workflow and output is really worth it, or if the congregation is better served with a high-quality standard-def workflow.