What instrument or program type? People may have more specific suggestions.
I have 2 kids that recorded for vocal. We did reserve a space and hire a pianist for each. My last kid’s teacher had access to a concert hall during the day she got for free, so definitely ask teachers, ensemble directors, etc if they might have ideas for a space espcially if you are trying to keep it affordable.
I think it is fine to video. But I highly recommend using a mic to pick up sound. My first kid had to record twice (long story). The recording with the better mic was SO much better than the other I really feel it could have affected results. I bought both my kids a blue yeti usb mic during covid and having a good quality mic just seems like a need for a music student anyway. Learning to merge video/sound is also a good skill to learn. But of course, you can hire all that out. And some local spaces here offer that as an option. We practiced filming/mering ahead of time at home and it went really smoothly. My kids use hit film express for editing, there is a free version.