Sounds like that app wants access to save something to your internal storage. This could be harmless things like favorites or settings, or websites and pictures for faster loading in the future. However, it could also start installing bullshit apps, ads, and other malware.

I would typically lean towards not accepting it, but I would allow it if I was completely certain I could trust the app. Major companies generally don’t want their official apps to destroy their customers phones, but some greedy app developer in his mom’s basement will happily take you for what you’re worth.