I was just curious why this isn't possible because it's clearly been considered based on the existence of them in the ISS for keeping the facility online. I'd appreciate some clarity on this. Thanks a lot.
The ISS does not have any sort of electric propulsion system. It uses ordinary chemical rockets, either on visiting vehicles or its own rockets which are refueled by such vehicles. (https://en.wikipedia.org/wiki/International_Space_Station#Orbit) Its solar panels are for running its experiments and other equipment.
There is only one rocket that uses lithium batteries as an energy source for its propulsion system (the Electron), and it only uses them to drive pumps for its kerosene and liquid oxygen propellants, which provide the energy for actually propelling the vehicle. When it's out of those, there's no point in recharging the batteries, there's nothing left to pump.
There are many spacecraft and satellites that use ion thrusters, and every single one of them uses solar power to run them...batteries alone could never store enough energy to get anything useful out of such thrusters. Also, while they use electrical power, they use it to accelerate propellant from an onboard tank. There is no "pure electric" thruster (excluding photon rockets, which have impractically high power requirements).
The majority of rockets (here meaning "launch vehicles") don't have much need for continued large amounts of electric power, and often don't need to operate for more than a few hours. Electricity is only used to power computers/guidance and controls, while bulk mechanical power is provided by compressed gas, gas generators, pyrotechnics, and turbopumps. All the fuel is burned in a short period getting into orbit, and perhaps a bit more after a restart to adjust orbit or eject into a geostationary transfer or interplanetary orbit, and after that the rocket stage is no longer useful.
For all existing or planned launch vehicles, propulsion is provided exclusively by chemical rockets which, with a few blazing exceptions, are not electrically powered (and the ones that are, such as the Electron, only use electricity to pump fuel into the engine).
(This is in contrast to satellites and probes of all kinds, which normally have solar panels or RTGs and have plenty of power needs. Additionally, manned spacecraft typically have fuel cells or solar panels to provide power -- but this power is still not normally used for propulsion.).
As a result, most launch vehicles don't benefit from having solar panels.
Deep space probes are (mostly) launched using two stage rockets. The first stage is not used after burnout and separation, recharging batteries of the first stage does not make sense. After separation of the first stage there is no connection of the space probe to the first stage.
The second stage is used to give the probe the neccessary speed and direction to the target of the probe. After burnout the second stage is not useful for the probe anymore and is separated from the probe. So recharging the second stage does not make sense too.
Solar panels attached to the rocket itself would only reduce the payload weight. So the possible space probe weight would be reduced.
But solar panels may deliver power to the space probe if the distance to the sun is not too far. During the flight through the atmosphere the probe and the solar panels may be damaged by atmospheric drag, therefore both are protected by a cover called fairing. As long as solar panels are under the fairing or not unfolded yet they generate no power.
To be not damaged by the acceleration of the second stage, solar panels of the probe are unfolded not before burnout and stage separation.