So Hugo is a static site generator and the server is for development purposes only, so I don’t think this feature makes a lot of sense 
That said, there are plenty of form solutions out there that allow you to embed forms inside static pages:
- Formstack
- Typeform
- Formspree (this seems like the best fit for your use case)
- Wufoo
- JotForm
- mailjet (Referenced below as well)
Note too that many on the forums use Netlify to host their static sites—this is my personal favorite as well. Even their free tier includes some really easy-to-set-up forms.