Can I install updates on an old iMac 7,1?

The highest you can go is osx 10.11, El Capitan:

http://updates-http.cdn-apple.com/2019/cert/061-41424-20191024-218af9ec-cf50-4516-9011-228c78eda3d2/InstallMacOSX.dmg

Your Mac originally came with 10.4, Tiger, and according to Apple page you will need to have updated to Snow Lepoard , 10.6.8, to be able to install this.

How to get old versions of macOS - Apple Support

For that you would need the install disks which are not available from Apple, but may be from independent retailers (however it's been a long time since they were current).