Yes, as long as you can provide episode download data via screenshots, that will be sufficient for most sponsors.
If your current podcast hosting platform or setup does not provide any analytics for your podcast, we'd recommend moving your show to a hosting platform that does. You can see a list of Apple's recommended podcast hosting platforms here.
We have also developed a new feature that allows for manual stats upload.
Go to Settings and select Manage Podcasts
Click the pencil icon (edit button)
Click on the Upload my stats tab
When uploading, please include a screenshot showing the results from your previous 5 episodes, including their exact listener counts and the average of those episodes. This information is necessary for approval.