Cron and hosting
A BIT ABOUT CRON FIRST
The great thing about cron is they need to be manually set. This means as a user you can control the tasks and frequency cron is running.
Now, let me give you a real idea of what cron is doing in this case.
You know when you're logged into osTicket and it brings your mail through and creates the tickets? The cron task basically runs the same script so you shouldn't see any change to the server.
A BIT ABOUT HOSTING COMPANIES AND CRON
When a website hosting company says they will not allow cron jobs or taskscheduler to run on their servers this raises a BIG red flag for me.
This tells me:
1. they don't know how to manage their servers and/or their customers correctly
2. they overload their servers to a point where running cron will cause problems with server load
3. they think it is still 1997 when the average RAM for a server was either 128Mb for an average server or 256Mb for a good server, this is what caused many server issues with cron, today the average is 2Gb and the performance issues generally just no longer exist
Under any 1 of these situations I would deem them to be a "bad host", a term I don't use often.
Regardless of their reason I would strongly advise you to seek an alternative host. There are many fantastic hosts out there that will meet the needs of most websites, including cron. Website hosting is a competitive world and those who do not provide the services needed and wanted by their customers have a limited life.
Technology is changing and hosts that cannot change or adapt with it have some serious problems.