How do I create a new support ticket?

You cannot create a ticket. You can go on chat with them and talk to them there. After the chat has ended the conversation will be emailed to you and you can respond to this email. Or you can email them to the same email support@shipstation.com directly instead going to the chat. 

https://help.shipstation.com/hc/en-us/requests/new

Try above while logged in.

It speaks volumes that a company we each pay $$$$ a year to has hidden their support ticket system so well that people have to ask for help in how to report a problem.

You'd think there'd be a link in 'my support cases' but no, which means at some point the following conversation happened at shipstation.

intern: 'my support cases' seems like the perfect place to have a link for customers to send a new ticket

boss: Oh no no no! If they can't report problems, then we don't have problems. Same reason we never update the status page when the site goes down!