I'm wondering if there are any depletion-mode transistors that allow a current through the gate/base/other control terminal?
I can't find any on the internet. I used the search function here.

 

As far as I know, all depletion mode transistors are of MOSFET type. These do not have gate current, like BJTs have. There would be current going in and out of the gate, but this is strictly for charging and discharging gate capacitance.

Any particular reason why you are looking for one?

 

Because I want to have a solenoid connected through it. I'll just use an optocoupler, like I have now.

 

A JFET is a depletion mode only device. A MOSFET can be either depletion mode or enhancement mode. Neither one requires appreciable current into or out of the gate except for turn on and turn off where it helps to charge and discharge the gate capacitence.

If you are using an optocoupler then you could also use a bipolar transitor especially if you don't need the isolation and are looking for faster response and more reliability. Did you know that the CTR of an optocoupler degrades with age?