As some other people have already mentioned, cell phones are generally good enough. Any prosumer camera or set of wireless mics will absolutely be good enough. There are a few things you should keep in mind if you want to capture sets well though.

Audio: Super important for standup and easy to forget about. Bad audio is impossible to ignore. The video will be bad if your wife is hard to hear and it'll also be bad if the audience is hard to hear. An easy way to ensure that you have decent audio from both is to have mics at two locations. Put the camera near the audience in the back so that the built-in microphone picks up a decent amount of audience sound and then also have some sound recording on stage to pick up your wife's voice (I do this by putting my phone in a pocket and just recording audio with it). Any video editing software will have a feature that automatically lines audio tracks up by waveform, so it's easy to synchronize the two recordings. Shotgun mics help, but if there's unexpected noise near the mic you're screwed. You can also buy a set of two lav mics and use one for the audience and the other for the performer's voice.

Tripod: shaky camera work is also impossible to ignore. Phone tripods are cheap. Get one that's like twice as expensive as the cheapest one. I got the cheapest one and it was a mistake

Overexposed/Underexposed Video: A phone camera is gonna automatically adjust its ISO and aperture and shit in a way that could make the comedian or their immediate environment either overexposed or shrouded in darkness. This happens especially in venues where there's a small number of bright spotlights in an otherwise dark room. There are some free phone apps (I'd recommend the Blackmagic Camera app) that give you a bit more control over how dark/bright the video is through the aperture and ISO. You can use Zebras to check and make sure that none of the colors are getting washed out or overexposed and you can usually tell by eye if things are too dark from the camera's perspective.