There is already a search field on the home page for FAQs/Knowledge base.
For a user to search tickets (theirs only, unless they are allowed to see other ticket for their organization) they have to log in.
I created a mod for something similar-ish. I wanted people to be able to see certain tickets so that I didn't get 100 people opening a ticket for the same problem (i.e. an internet outage, etc). The article for how to do it is here:
http://tmib.net/add-a-client-side-open-ticket-list-to-osticket-1-10/
note: it will take some customization so you will need to be familiar with PHP to make it work in a less vanilla way than the tutorial. Also this was written for 1.10 so the instructions might take a little trial and error to work on 1.11. [But I am running it on 1.11 currently]