I was wondering, is there a way where I can update it without using an external disc?
If you mean the iMac does not have an internal optical drive capable of reading DVDs, the only option requires another Mac with a DVD-capable drive and FireWire ports. Your iMac G3 will also require FireWire ports. Not all G3 iMacs have Firewire. FW started with the Summer 2000 models with 400mHz and faster processors.
With FireWire, you can use FireWire Target Disk mode to install from a retail OS install DVD in another Mac. If you supplemental info info tells us if you have another Mac with a DVD reader and the FW ports, we can walk you through the process.
If the iMac has a tray-loading drive, I'd forget it. They are very limited not only in the OS max (10.3.9) but also in RAM capability. Starting with the slot-loading iMac G3s, the max RAM increased to 1 GB (2 X 512MB modlules). Tiger likes 1GB RAM.
Understand that pre-Intel Macs are getting harder to use. Even with 10.4.11, your choice of browsers is limited (TenFourFox is the most viable option) and the G3s rudimentary video hardware means web browsing will be choppy at best.
If you are in the USA, you can try calling the Apple Store (1-800-MY-APPLE) for an OS disk. As recently as Jan 20, someone posted here that they were able to buy Tiger by calling. It was not clear whether the disks were replacement system install/restore disks or full retail install, but it's a free call.
When you do "About this Mac" to check teh OS verion, also look at the processor speed and tell us what that says. That can help with recommendations for an upgrade path.