Lets say I were to donate my organs to a person of my choice. Even if they didn't need the transplant right now, is it possible to still remove my organs and keep them on ice just in case anything were to ever happen to them?

I'm not a medical professional, but as far as I know organs have to be transplanted rather quickly after their removal from the donor. They are now able to keep kidneys in good condition for a day or so, in order to transport them elsewhere for implantation, but I don't think organs can be preserved for a long time.
    Fr. Pat

The shelf life for tissues for human transplantation differs significantly depending on which tissue, for what purpose. For the long term, you could direct your heirs to remove your bone marrow and preserve your bones for potential use in the far future, but the expense and emotional load seems excessive for a very low probability of utility to any of your heirs or descendants. Directing your heirs to donate all your tissues appropriate for use upon your demise saves multiple recipients immediately, improving society incrementally by increasing useful lifetimes, as well as incrementally expanding positive progressive altruism as a societal norm. Much surer positive impact on your loved ones, immediately and in the long term. Best wishes.