I had the exact same problem this afternoon trying to install StockDio on the latest version of WordPress using both Chrome and Microsoft Edge browsers. As the original poster wrote, the settings kept refreshing and I was unable to enter my app key.
I found the solution, however, which was to disable Jetpack while updating the settings, then reactivating it again afterwards. I don’t know why Jetpack has this effect on the StockDio settings page, but it did the trick…
