Firstly, understand that PWA or Progressive Web App by definition must support all Browsers and should be Progressive in nature, i.e., depending upon the capabilities of a browser the app must adapt and choose what features to leverage in order to elate the user.
A quick analogy:
A Shark in a fish tank will grow 7 Inch but in the Ocean it will grow
7+ Feet.
Your App is like a SHARK!
Fish tank/Ocean is like your Browser !!
Resources of Fish tank/Ocean are your Browser Features
Be it Fish tank or Ocean, a shark is still a shark (It looks like one,
it preys). What differs is the scale/size/capacity/performance (e.g.,
Sharks in the Ocean have larger teeth and prey larger fish).
Applying the analogy, Your app design must not SOLELY depend upon Database/Storage (or any other Browser feature) to render itself, but at the same time, it should be able to use them if supported by the Browser in order to provide better User experiences. (Meaning, PWAs should work even if say localStorage is not supported by some weird Browser - too much to ask !)
Elaborating further, Your app could try to use IndexedDB on a Browser, if not, provide an appropriate fall back mechanism (fall back need NOT essentially be Local storage at all!, but perhaps a simple render). IndexedDB is used to store large amounts of data which require rich query abilities, whereas, localStorage is similar to sessionStorage that persists even after you close the window, usually meant for storing some bunch of key/value pairs. Trying to use one over other would be incorrect - they are meant for different types of datasets/usecases.
In short, in PWA context, it is WRONG to think "IndexedDB or Local
Storage?", but to consider what experiences you want to provide your
users based on limiting Browser features at your disposal.
Good luck with your PWA !!!
Interesting read on available storage options on different browsers
https://developers.google.com/web/fundamentals/instant-and-offline/web-storage/#comparison