Having seen the Doctor seemingly need to use his hands to manipulate the sonic screwdriver, I was wondering if the Doctor could instead utilize it via just his mind or some other form of remote control to use the sonic instead. So he isn't so vulnerable if somehow denied the ability to use it manually. Or does he strictly have to use his hands to use the sonic?
It has a psychic interface, but there's been no seen cases of it being used while not in the user's hand.
In Let's Kill Hitler, The Doctor explains that the screwdriver operates by the user simply thinking of the function needed and pressing the button. This may be a new function, as there's been cases of The Doctor fiddling with the device to get it to perform certain functions - he tells Rose in The Doctor Dances that a particular numbered setting cuts and/or reconnects barbed wire, and in Robot, The Doctor attached something to the Sonic to make "a miniature sonic lance".
The sonic probably cannot be used hands-free.
In "Deep Breath", the Doctor and Clara are captured. The Doctor drops the sonic onto Clara's feet, and she has to carefully pass it up to his lap in order to operate it. He was probably unable to use it hands-free, or they would not have needed to do this.
It depends on which sonic screwdriver you are reffering too, because each sonic screwdriver is not completely the same with some screwdrivers functioning on different wavelengths as mentioned in "Dr Who Demons of the Punjab" the doctor had to find the correct wavelength for what she needed, so with some screwdrivers maybe the doctor could but maybe not.