Can I update the software past SnowLeopard?

Your system supports upto High Sierra (10.13.x) Frankly, I would stick with Sierra as High Sierra alters the file system from HFS+ to APFS. HFS plays better on older SATA based systems.

So with that here’s the location to get the needed OS’s How to get old versions of macOS. Get a 32GB USB thumb drive format it to GUID with a journaled file system then copy over the installer to it. Now follow this to covert it to a bootable OS installer How to create a bootable macOS Sierra installer drive

Once done you can upgrade Safari to the newer release using the App Store.