SQL Server is a A.C.I.D compliant relational database platform, and transaction logs are a critical aspect of this component. Without a transaction log, SQL Server could not ensure any of those attributes. SQL Server wasn't designed to be a 'file store', it was designed to be a data store. Essentially you guys might be using the wrong tool for the job.
If your transaction logs are as big as your actual data files you might not be truncating the data within the log files, or there's a lot of optimization the app layer needs in finishing old transactions. Are you in the SIMPLE recovery mode?
Yes, I have been in shops that use SQL Server to store binary data or in this case, .zip files, it works but there's issues as well. Typically they go with the VARCHAR(MAX) data type, which could let you fill up the entire hard drive with 1 row of data, and it happened before.
If you have to, look at FILESTREAM or the 2014 in memory tables, although I don't think in memory tables work with BLOB object types yet. Neither of those will help your transaction log issue though.