Its technically possible, except Airtable’s attachment type fields, when updated through the Custom App API, must be fed a URL of the attachment to upload.
That means users won’t be picking something from their file system, at least not technically. Your app would have to upload the file from their file system to somewhere on the internet (i.e. Google Drive, Dropbox, etc.), and then pass that URL to the attachment field so that the file gets added to the base.
@jordanscott
Related to this and the AMA API Webinar question about adding support for uploading attachments directly, are there any plans to add support for this soon instead of only URL’s?