Older CiviCRM that does NOT require TRIGGER privileges?
I'd recommend approaching the problem the other way round: find a hoster that works with the latest versions of Civi rather than find a version of Civi that works with your current hoster.  You will be missing out on many new features and security fixes if you choose to run an old version.  It will only bite you later.

Is there any reason not to move hoster?  As ErikH points out, there are several specialist CiviCRM hosting services - but not all are equal.  See eg How to evaluate hosted CiviCrm enviroments