There is a daily lottery two days before the hike date, and there is no walk-up permit process that I know of. Although when I stood in line for a walk-up overnight permit, there were several people looking to pick up one of those to do the day hike.
If you repeatedly enter the daily lottery, you will eventually be successful, but if they charge the $10 to enter each day, that can cost $$$.
You can read up on the daily lottery here
. That web page has links to the best success rates for the daily lottery last year. Mid-week days show the best success rates.
And then there is always the no-permit-required period, from about mid-October to mid-May, when the cables are "down". No permits required at all. But if you are a bit intimidated by heights and such, that would not be a good idea. If you talk to park personnel, they always try to talk people out of doing the cables-down hike, but you will find a dozen people practically any day doing it when weather is good. Here's a link: Half Dome with Cables Down