It's possible, but I can't find any existing solutions that solve this need.
The best possible data for this will come from the big search engine's APIs (e.g., Google, Bing) due their global reach and massive data storage capabilities.
If you truly want to self-host something, I'd suggest looking into setting up a simple pipeline from one of those (or multiple) APIs to a self-hosted data store (e.g., Elastic, Postgres) and write up some simple scripts that will let you search at-will like you would with Google Trends.
