All is well now.  I had to contact our hosting provider (I didn't have the grants to make any changes) and we eventually got it straightened out.  Here is what they told me:
One of our administrators looked into this further. They noticed that this is a problem with iconv.so, which is a FreeBSD port that Ruby was looking for. We already have that port installed at /usr/local/lib/ruby/1.8/i386-freebsd7/, but Ruby was looking for it in /usr/local/lib/ruby/1.8/i386-freebsd7.2/.

He copied that port over to the location that Ruby was looking for, and was able to run "./s3sync.rb --help" within your account. You may want to try running this again now.

I am having a similar problem:

S3encoder.rb:17:in `require': no such file to load -- iconv (LoadError)
   from /usr/home/boatrv/backup/s3sync/S3encoder.rb:17:in `<top (required)>'
   from /usr/home/boatrv/backup/s3sync/s3try.rb:29:in `require'
   from /usr/home/boatrv/backup/s3sync/s3try.rb:29:in `<module:S3sync>'
   from /usr/home/boatrv/backup/s3sync/s3try.rb:10:in `<top (required)>'
   from s3sync.rb:25:in `require'
   from s3sync.rb:25:in `<module:S3sync>'
   from s3sync.rb:11:in `<main>'

I am using the current version of Ruby (1.9.1).  Can anyone offer any insight?