You're actually asking three quite separate questions there... 1. where does the energy come from? (eg. do they need a dyson sphere?), 2. how does the energy get into your picotech... things? devices? mechanisms? (John Meaney called his femtotech "engineered psuedatoms" in his Nulapeiron books, what are you calling yours?) and 3. how do your devices use that stored energy to interact with the world?
I'm going to ignore (1), because it is less interesting than the other question and as it isn't at all clear how much energy your technosphere needs it isn't really something that I can usefully answer.
(2) is slightly more interesting.
Devices of that scale can store energy in the form of metastable states called nuclear isomers. They're mostly associated with radioactivity, and the interesting possibility of induced gamma emission which could be used to make a super-high energy density battery or a powerful (but non-fission/fusion) explosive. If you can rejiggle internal atomic states at will, working with nuclear isomers would be child's play. Each individual picotech device would be "charged up", undergoing isomeric transition into a long-lived metastable state, with the energy released at some point in the future to enable some kind of atomic-scale action, whatever that might be.
Charging such a device is awkward, because it require interacting with the nucleus of your custom atoms. This basically requires either bouncing particles off the nucleus, or bombarding it with suitably short wavelength gamma rays. This is something that you might do inside a nuclear reactor rather than something you'd do to devices floating around in your atmosphere.
And that kinda segues into (3). I posit that you don't really want to be using pico-scale devices for everyday tasks, because everyday tasks don't generally require very much nuclear chemistry, and things that do involve nuclear chemistry tend to be hazardous to be around and you don't want them wafting around your atmosphere or sea. You use this technology for making exotic bulk materials, or very specialist nuclear manipulation tricks, but for everything else it is either too powerful and dangerous (like using a railgun to crack a walnut) or simply not large scale enough.
Most things you care about are probably made of quite large molecules. This probably includes your good self. Molecular-scale interactions are demonstrably useful. Moreover, molecular scale systems already demonstrate useful ways to absorb and release energy without recourse to nuclear reactions... consider stuff like chlorophyll that operates on regular light, which is generally quite safe. Other structures could absorb or emit other wavelengths, allowing for radio control, communication and power transmission. Everything is easier at that scale, and things are generally a lot more useful, too. Sure, your nanosystems might be made of materials created by nuclear engineering, but it is the nanometer-scale chemical properties of those materials that matter most.
Honestly though, I'd conserve your details here as much as possible. Clarke's third law definitely applies. You aren't a godlike superintelligence, and you don't find many of them contributing to stack exchange. Anything anyone can imagine for the functioning of picotech is probably wrong, and has a good chance of being badly wrong.
If you don't go into detail, you can't be wrong. Use handwaves to convey mystery. If your godlike superintelligences aren't mysterious and incomprehensible to some degree, you might be doing it wrong.
