The few I had a pleasure of using have a really good browser and mobile app support. The only problem I ran into is that sometimes mobile apps refuse to run on rooted Android devices.
Raiffeisen in Austria is/was? one of the stragglers, but I haven't dealt with them in 2 years or so - a company I worked for was stuck with it, but I noped out for Erste Sparkasse ASAP, and they let you do everything in browser. (Since 2011 at least.)
Raiffeisen in Russia let one do everything in a browser on Linux since 2008 for personal accounts. Possibly earlier, I did not have an account with them back then. For business accounts, it's since February 2016.
