is there a way to make the website search bar "smarter" without reaching out to a third-party company ?

If it's your own website, I feel an integration with algolia which uses elastic searches works like a charm.

its an SCA website

Depends what you mean by ‘smarter’. You can configure which fields are searched/indexed, which ones are prioritised, and how words are matched against them (eg fuzzy, exact, starts with, etc)

i search for headset on my site, and i see results. When i search for head sets - nothing comes up.

Then you either need to set up fuzzy matching and/or synonyms