I get better response times here in OT!

What are their SLAs?
Service Level Agreements.
Most will include things such as "initial response within ### hours" and then specific time ranges depending on your tier (gold level support).
A quick automated reply with your case number often is all they need to do for 1-3 days.
Is there a "customer portal" where you can log in and check the status of your ticket? View their updates and such?
If it is an urgent/important issue, I would request updates daily from vendor support. also would request that they include their name in the update.
Sometimes, (especially in summer and around holidays), a staff member that is assigned to your ticket won't pass it on to another to continue/complete, they will just disappear and then return, and if there isn't a notification system baked into their ticketing system, they may forget all about your ticket.