The XDA site/forum is the place to start your investigations... but it's a bit techie. Putting a new OS (or version of) on your phone is not always straightforward and you do risk knackering your phone entirely. I've upgraded a few phones/tablets now (because, well, just because and I'm a bit geeky like that!) but I've had a few close calls where I've thought I've buggered the device.
On top of that, sticking a newer version on the phone may not be the answer. Even IF there is a later version available AND it works, it could well grind your phone a halt, making it practically unusable. There are hardware limitations as to what version OS a device can happily, responsively run.
In short, if you're otherwise happy with the phone then I'd seriously consider living without that one app. OR - you might be able to downgrade that app and 'freeze' it - stop it being automatically updated. That's not always possible, it depends on the app.
It's of no help I know but I'm in the same boat as you with an old tablet I have - an LG Pad 7, it's running version 4 like your phone is. There are no manufacturer supplied updates available for it, it long since having been ceased. Looking at XDA, I can find no upgrade candidate for it, not that looks 'safe' at any rate. I also have a Sonos sound system and the latest Sonos android app will not run on android 7 - and because of the Sonos side software changes, the version 7 compatible Sonos app wouldn't work with the system anway. I've simply had to live without that app on that tablet. Damned annoying but sometimes you do have to accept the end of the road (for all practical purposes anyway)