Yes, many CRMs (Salesforce for example) can harvest keyword (search_term) data from your web form if you use UTM tracking codes.
The best and easiest solution, though, is CallRail. It’ll automatically grab all the information from your webform (and calls) and include the keyword data for you. And it’s only $85/month for full call and form tracking.
The nice thing about this is you don’t need to use forms that support hidden fields, which can otherwise mean a redesign.