I'm sure that you explained to them some problems in the previous spreadsheet methods,
there can be:
1- data redundancy
users can enter one data many times
2- there isn't any security for don't losting data
users can delete data suddenly
3- cuncurrecy with working is impossible
when two users download spreadsheet and change it , how this data can be merged?
4- there isn't any access roles there
maybe some data shouldn't to be show to special users, this can not be done with spreadsheet
and many other acceptable reasons, I think the first reason mere can be acceptable to move to web site method.