So I stopped at a place where I had an errand to do, and when I came out (maybe half an hour later?) I went to the Tesla Widget. The car was asleep so I hit the refresh button and it started going around and around, and then finally gave up.
My phone will not wake up my car if it's asleep.
At my next stop I went to the widget right when I got out of the car, and the buttons were there and I was able to lock the car from the widget. Didn't bother trying to open the frunk but the button was available.
So, yes, the widget has a frunk button, but when I really need to use it the car will probably be asleep and the app/widget will not connect. The car unlocks when I pull on the handle, so the BT connection is there. And I have a cell connection on the phone and data turned on for the Tesla app. But the car seems to turn off its cell connection when it goes to sleep and that's it until I open the door. It's weird because others don't seem to have this problem. And I guess the app/widget won't use the BT connection for anything but unlocking from the door handle.
All-around poor design, if you ask me. I can live with it to be able to drive the best car on the road.