It's currently impossible to start a bounty without the mouse. Trying to tab to the "start a bounty" skips right over it; going from "add a comment" to the link to the question in the "share a link" thing.
You should be able to tab to the button, press enter, select an amount using the arrow keys, continue, select a reason using either the arrow keys or numbers, continue, write a custom message, and start the bounty.
If you're confused as to why it's important that things be accessible without the mouse, please read up on keyboard accessibility. While I thankfully don't have any physical problems with using a mouse, I'm using my time with a broken mouse to give accessibility feedback so that people who need it can get a better site experience.