Well, guess what?
You own a nearly 9-year old iPad Mini forever stuck on iOS 9.3.5.
It looks like DropBox developers have finally dropped support for their older app on old iOS devices.
I own old and new iOS devices.
My 10-year old, 2011 iPad 2 is exhibiting this same behaviour.
I tried deleting and re-installing the older app to no avail.
This older DropBox app no longer works for 9-10-year old iOS devices,,running iOS 9 and, probably, iOS 10, as well.
This is going to happen more and more with may Web-based apps on older or very old Apple iOS devices.
Sorry.
Apple ended support for all 8-10-year old iOS devices well over 4 years, ago, in 2016.
Third party app developers abandoned and stopped supporting these 8-10-year old iOS devices nearly 3 years, ago, in 2018.
There are, virtually, NO MORE third party apps available from the iOS App Store, any longer!
Third party compatible apps for such old devices are few and far between, now, in 2021.
AND, that well over 4-year old default iOS Safari Web browser is too old, not secure and not safe to use to access the Internet, any longer.
There are virtually NO MORE third party Web browsers left for 8-10 year old iPads!
Available and more popular third party Web browser choices are pretty much GONE, now.
If you must use these old iPad models, here is what is left for still updated Web browsers.
Pickins’ are slim, now!
ALook Web browser ( A fairly new Web browser primarily designed for old iOS device versions, but is still available to newer iOS/iPadOS devices, as well )
Dolphin Web browser
Maxthon Web browser
TOR private browser
TOR + VPN private browser
Turbo VPN private browser
This is pretty much it for iOS 9/10 compatible, third party Web browsers today.
Simply put, these 9-10 year old iPads are too old and obsolete now.
Sorry, but this is the reality for ALL 8-10-year old AND OLDER iOS devices now. Especially going into 2021!
You will need to use a Web browser to access the DropBox Website, now, and sign into your DropBox account, using your DropBox credentials and access your DropBox content from a Web browser now.
Most Web-based apps are mainly just portals for the actual Websites, anyways.
2021 now!
If you need/want an iPad with an up to date iPadOS with access to current, updated apps, then you really need to seriously consider a purchase of a new iPad model OR a much, MUCH “newer” and less expensive, refurbished or used iPad models, from better known, trusted Apple product/device retailers/resellers, locally or online, on the Web/Internet, capable of running the latest iPadOS versions.
There are PLENTY of newer, older iPad models that can run the latest, current versions of iPadOS.
I am sorry that you had to learn all of this, but this does not change any of the facts regarding these very old iOS devices.
Sorry & Best of Luck to You!