This sounds interesting, but it’s not possible with Mixxx right now.
Sending keystrokes via MIDI is not implemented, though that could be a nice feature.
As easy as it may sound, the low-level typewriter you suggest is a complex feature with rather limited use cases so I doubt anyone would spend time on implementing that. Plus it had to not conflict with all standart functions of the search box.
Also, the search box will get new features in Mixxx 2.4: up/down keys scroll through all previous search terms, press Enter to confirm. We still have to figure out if and how to make those queries persist across restarts. If that works you could simply prepare the queries you need most and reload them later on with Up/Down (turn encoder cw/ccw) and Enter (encoder press).
For now, is a (small) touch screen an option for you? Most OS (when properly configured) would enable you could tap into the searchbar to bring up a on-screen keyboard.
Another idea is to build an additional tiny “controller” which is a poject of it’s own, but maybe sparks other ideas: there are tiny Arduino microboards (1"x1") that can be programmed to act as USB keyboard. Add a turn/press encoder and transate its input signal to key presses or key combos. This could allow you to bring up any on-screen keyboard, navigate the key rows with left/right turns and confirm with press, and also hide the keyboard. You could also code more complex behaviour like long presses, press-and-turn etc.
If you like that route but don’t want to built it yourself then there are ready-made solutions like the Surface Dial, or other ‘USB volume controller’