I have a bluetooth keyboard that I sometimes use with my phone when traveling and don't want to deal with the tiny phone virtual keyboard. I had never thought to use it with the Roku app on the phone so I just gave it a try. With the Roku app in keyboard mode it worked just fine. I tried it out doing a search from the Roku home page and my bluetooth key presses transferred to the Roku search as one might expect.
The fly in the ointment, as @andyross indicated, is that not all Roku channel app developers have incorporated support for the Roku remote app keyboard. My Prime video does not accept Roku app keyboard entry, for instance, while Hulu does. For apps without Roku app keyboard support, my bluetooth key presses get to the Roku app, but go no further.
Roku Community Streaming Expert I am not a Roku employee, only a user like you. Please, no support questions via private message -- post them publicly to the Community where others may benefit as well. If this post solves your problem please help others find this answer by clicking "Accept as Solution.".
Streaming Stick 4K+ 3821 | Streaming Stick+ 3810 | TCL Roku TV 43S245/C107X