They’ve shown The Doctor charge the TARDIS on multiple occasions, but I don’t recall them ever showing the sonic screwdriver being charged or anything. Maybe it's because The Doctor goes through them too quickly?
There's a charger slot for the sonic in the TARDIS console.
At least in the Eleventh and Twelfth Doctors' TARDISes.
In the 2013 Christmas special "Time of the Doctor" (emphasis mine):
DOCTOR: A question only I could answer. A truth field to make sure I'm not lying. If I give my name, they'll know they've found the right place and that it's safe to come through.
CLARA: The Time Lords? Okay, so what then? If you answer the question and they come back, what happens?
(The Doctor gives Clara a short round device.)
DOCTOR: Er, you need to take this to the Tardis and put it in the charger slot for the sonic.
DOCTOR: Hell. All hell, that's what happens if the Time Lords come back. There's half a universe up there already, waiting to open fire. Now please, go to the Tardis and just do as I say.
And in the Series 9 finale "Hell Bent", we see the sonic materialise in its charger slot before flying into the Doctor's hand:
It's powered by "batteries" (also referred to as "power cells"). Occasionally these run down and need to be recharged.
‘Is it just me,’ said Flowers, ‘or is the light getting fainter?’ She shook the sonic screwdriver, clamped tight in her clammy hand.
The Doctor rolled back on to his stomach. ‘Battery’s running down.
"The Doctor keeps the tool on him for as long as power cells allow between recharges" - "Charging cells [depicted]"
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