Problem:

I am trying to speed up the search of my view, and am wondering if it is possible to implement such a jQuery snippet of code to tell the database to begin searching through the database as soon as a user leaves the input field.

Solution:

There isn't such kind of built-in feature within Toolset plugins, you might consider custom codes, for example, when event "onfocusout" is triggered, you can submit the search form with jQuery submit(), see jQuery document:

https://api.jquery.com/submit/

Relevant Documentation: