There is no fee to have your refund direct deposited to your bank account. You only pay a fee when you choose the Refund Processing Service (to pay TurboTax fees from your federal refund) option. That is disclosed in the agreement that is presented to you to read and sign before filing.
What you do pay for is the TurboTax software, any state returns you add or any other add-ons that you add before filing.
If you chose to pay fees with federal refund, yes, it goes to Santa Barbara Tax Products Group first. And if you checked the status of your refund from the IRS Where's my Refund? page and it says that your refund was sent to your bank, but TPG hasn't received and released your refund please be aware that the IRS may indicate funds are available days before they are released to TPG and other banks industry-wide.
You can go here Check the status of your Refund Transfer Status with TPG to check status of it.
You can also find their contact information here Santa Barbara TPG Contact Information and you are correct, there is no chat, but there are phone numbers listed.
They also provide details as to why they would hold your funds and how to release any holds on your refund here How to verify my information with Santa Barbara Tax Products Group (TPG)
