Hi Peter,
It depends on your remote storage method.
S3, Dropbox, CloudFiles and Google Drive all require Curl. FTP, SFTP, SCP and WebDAV do not require Curl.
Really, though, I’d recommend you get a different web hosting company. If they disable Curl, but do not disable all networking, then they are not raising security. What is done in Curl can be done in pure PHP (just with a lot of programmer effort). It sounds like that is your situation – you want something that reimplements Curl functionality in pure PHP. But, if your host allows the same operations as Curl performs to be performed by other means, then they have not really raised security, only annoyance!
David