There is a mappable sprint key for a mouse & keyboard, but not a controller. On the current Console Editions, (such as PS4 and Xbox 360 Editions) there is a remappable sprint button, yet it is absent in the Bedrock Edition. I don't feel there's a need for a "to reproduce" section of this report due to the obvious way that it is "reproduced."
MCPE-25020
No Sprint Button for Controllers
Resolved
Fixed
2
(Unassigned)
Community Consensus
Tablet - Android - Nvidia Shield Tablet
Android 7.0
1.1.5.0, 1.2.0.15, 1.2.0.22
1.2.5.52
Retrieved
This is even an issue with the new Xbox 1.2 beta (build 1.2.0.20).