Is it permissible to take a percentage as commission from an accounting software company in exchange for training employees at the institution where the questioner works, noting that the institution is unaware of this percentage, and the software company is unaware that the questioner works for the client?

You did well by not taking the commission, as you are an agent for the institution, and it is not permissible for an agent to take a commission without the permission of his principal.

As for training employees for a percentage from the company that owns the program, there is no harm in that. However, it is not permissible for this to occur during your institution's working hours except with its permission. We advise you to avoid anything that raises suspicions about you.