There has been a few attempts at creating a distributed search engine. You can read about them in Wikipedia https://en.wikipedia.org/wiki/Distributed_search_engine
None of them has indexed a very large amount of sites however. One issue with a distributed search engine is that it's pretty useless until it has a lot of data and until its useful, not a lot of people want to run it to index more sites.
Perhaps a solution to this problem could be to create it as a distributed autonomous organization which can pay people for indexing and crawling sites, storing indexed data etc. It could perhaps do this with its own coin, new coins might be created as new sites are indexed or something like that.