Apple provides System 7.5.3 on their website as a free download for users of older Macs. 7.5.3 should work well on the LCIII unless you only have 4MB RAM, in which case you'd either want to upgrade the RAM or find System 7.1.
System 7.1 is not free and never has been. The LCIII, like all models released after System 7.1 came out that are capable of running it, needs an enabler to run. System Enabler 003 is the one you'd need for an LCIII; it must be installed on your 7.1 "Install" diskette, should be on your "Disk Tools" diskette, and must be installed on your hard drive when you install System 7. (If you have it on your "Install" diskette it should go on without a problem). The enabler can be downloaded from Apple's site and placed on a 7.1 diskette, which you'll have to track down on eBay or purchase from a used Mac seller such as B&R Computer.
You're best off using 7.5.3, which has more features. A 32MB RAM SIMM will help maximize your LCIII's usefulness, so try to find one if you don't have one installed already. (Since you can't boot it up right now, pop the lid and see what is in the lone SIMM slot to check your RAM configuration).
Keep in mind that you cannot use floppy disks with Snow Leopard, so make sure you're using an older Mac if you take advantage of Apple's free downloads.