I think it is possible to renew your hosting after the expiry date has elapsed. There also will be an Auto-Renew Grace Period (0-45 days) and followed by an additional 30-day redemption Grace Period. It is after the Redemption Grace Period has elapsed when you will no longer be able to renew your domain name.
This need not worry you.
1. Download all your databases and make sure you have a local copy saved up on your computer.
2. If this is something you are not able to do simply contact your hosting provider, and have them share with you a link where you can download your important website backups.
3. Upon deciding on what hosting service you'll be using next, share these with them and they'll be able to restore everything back to normal.
Yes it is very much possible. Just make sure from the time the site expires and the time you'll be doing a renewal not too much time has not elapsed. If you wait too long then there's a pretty good chance the hosting company may end up deleting your website data. So between 1 week or two is good. Longer than that you may encounter issues unless you talk to them.