Have an older Peerless faucet with two clear plastic on/off knobs and the hot water side is leaking. Turned water supply off and took handle off on the hot water side and there appears to be a plastic cartridge inside with an O ring on the outside of the cartridge and smaller one inside. Will just replacing the O rings solve my problem or do I need to replace the whole cartridge? Or, is this even the problem as to why the faucet is even leaking?

Is it leaking from the spout or from the stem itself, around the handle ?

Yes, you could replace O rings . Home Depot sells repair kits. However, it is a good practice to replace entire stem. It is not expensive item at all and will take you less time then replacing O rings.

We always replace both, hot and cold, stems when doing service on faucets. This way, stems will "feel" the same and will start deteriorating at the same, exact, time.