[refactor] search.js: use custom auto completion implementation

The previously used library is unmaintained for 6 years now [1] and the solution
had know issues [2][3]

[1] https://github.com/searxng/searxng/pull/4284#discussion_r1954493434
[2] https://github.com/searxng/searxng/pull/4318#issuecomment-2731576657
[3] https://github.com/privau/searxng/issues/56

client/simple/package.json

@@ -34,8 +34,5 @@
     "vite-plugin-stylelint": "^6.0.0",
     "webpack": "^5.99.8",
     "webpack-cli": "^6.0.1"
-  },
-  "dependencies": {
-    "autocomplete-js": "^2.7.1"
   }
 }