For a free Netlify integration, you cannot create a 2nd crawler. You can only have one valid domain at any given time.
If you are using a Docs site, you may be eligible for Docsearch. Check if you conform with this checklist, and you can apply for access to the crawler via Docsearch.
If you aren't compliant with that checklist, you can also try running your own using this guide.
The free Netlify plan, doesn't include technical web support from the Support team.