Is there a way to get an "email digest" once a week with submissions instead of instant emails?

You can signup for Daily Digest service to receive daily emails containing all of your submissions. Here's the link: http://digest.jotform.io/

While here's the full guide for your reference: https://www.jotform.com/blog/daily-digest-email-service

You can set up as many daily digest emails as you wish and even customize it to be sent either daily, week, or monthly.