Often we migrate content from the existing site to a new sandbox, which means we basically grab all your existing content manually. However, in some cases you may have hidden files or old archived files on your server that aren't visible to us when we do the migration.
A simple way for us to get everything would be to grab the files via FTP. If you're able to give us FTP access / login credentials so that we can see what's there and download a backup of your current sites' content, we'll know for sure that we have all your resources at hand. If your current host is the only one with the credentials and you don't want to let them know you're going with another vendor yet, it can always wait until later in the process. We don't need a lot of lead time to grab a backup, even a few days will do (although of course we'd like a bit more time if possible).