Seems Mojo is pretty different from CGI in passing form parameters with GET.

Not really, CGI.pm also has an OO interface with a ->param method. Everything else is different, though :-)

How to make make it do partial matching (and possibly case insensitive)

That's more of an SQL question; it will depend on the database you're using. The most important thing is that if you build SQL queries yourself, you need to be very careful to quote everything correctly, usually using either placeholders or DBI's ->quote method, which in my code above you can access via $c->sqlite->db->dbh->quote(...) (or app->sqlite->...).

SQLite and DBD::SQLite supports an extension called FTS5 that might be helpful to you. For example, if in my code you replace the CREATE TABLE with this:

CREATE VIRTUAL TABLE contacts USING fts5(name, department);
INSERT INTO contacts (name,department) VALUES ('John Doe','Perl');
INSERT INTO contacts (name,department) VALUES ('Larry Wall','Perl');
INSERT INTO contacts (name,department) VALUES ('Gloria Wall','Perl');
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And the $c->stash( rows => ... value with this:

$c->sqlite->db->query(
  q{ SELECT rowid,name,department FROM contacts WHERE name MATCH ? },
  $c->stash('query') )->arrays );
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Then you get the full power of the FTS5 extension, e.g. your query can be john OR larry, g*, etc. and it'll just work.

Edit: Typo fix.