Spoilers: There is a phone number. For some things.
I frequent the Roku support boards and the question about how to call Roku to speak to customer service comes up often. People have told again and again that there is no customer service number for Roku, but they keep insisting they want a number to call.
Why are they told there was no customer service number? Because there was no customer service number.
But now there is. Hallelujah.
According to the support forums, Roku restarted telephone support for customer service a few weeks ago: (816) 272-8106. It's only for addresses account/billing issues (option 1) , and first time setup/activation issues (option 2). Nothing more. But, that's better than nothing.
If you're new to Roku, this might be helpful. I've not called it, nor am I likely to. But if you need it, and if calling makes your Streaming Life easier, there it is.