Thanks for your enquiry.
Certainly unusual not to provide ZipArchive class but BackupBuddy does adapt to that.
For taking backups BackupBuddy will use the "best" available method and this is deemed to be the native zip command as available through the PHP exec() function. In your case the other method it could use would be pclzip class (which we call Compatibility Mode) which is a library that ships as part of WordPress so _should_ always be present. From the information from your host they must support the native zip command through the PHP exec() function so BackupBuddy will use that as first choice so no problem there.
Additionally, for doing what we call an Integrity Check on produced backup files BackupBuddy needs to open a zip archive file that has been created previously (in your case by the native zip command) and for this it would use either ZipArchive or pclzip (so in your case it would have to use pclzip). We have not come across any specific problems with this in the past (in fact sometimes pclzip is better than ZipArchive in this respect) but as with everything software based there are limitations based on such things as memory availability and execution timeouts in relation to the size of the backups you might have and what their content might be (e.g., large backup because of a few large files vs large backup because of a large number of smaller files, etc.) The server performance can have an impact in some cases. I only mention this for information because unless your server has particularly tight constraints or is particularly slow you are very unlikely to have any issue and if you do your host support should be able to help in resolving it.
As for other plugins that may require to use ZipArchive I cannot say - if they depend upon ZipArchive then it will be an issue whereas if, like BackupBuddy, they can adapt and use other available methods then I guess it should not be an issue. If you have any concerns about any specific plugin then probably best to contact the authors of that plugin.
I hope this information will prove helpful to you and please let us know if any further clarification is required, thanks.