So you'd have to download the json of all the forum members usernames in one request?
That'd be pretty slow. Then doing a search through the json data every time someone presses a key would also be slow and intensive. That technique would also mean having to write server side code.
You could extend this idea by indexing any viewed threads against thread id's, then have a quick search to quickly jump to threads you've previously viewed by quickly typing part of the name of it.
If there are 500 members with 32 character usernames, that's 15KB of data - not slow to download.
Whether it's slow+intensive to use that with JSON vs SQLite is my question really - is there enough benefit from SQLite to make it a feature that can only be utilised by people who have installed extra software?