I don't believe that Airtable has plans to provide such a service, although that would be cool.
Your image would need to be temporarily public while uploading to Airtable via Airtable's REST API, but then you could delete the image afterwards.
I usually use Google Drive, Filestack, or Cloudinary or these purposes, and I automate all of this using Make's automations & integrations. (I've found that Dropbox's and Box's APIs can be challenging to work with, so I typically avoid them.)
Hope this helps! If you’d like to hire an expert Airtable consultant to help you with anything Airtable-related, please feel free to contact me through my website: Airtable consultant — ScottWorld