where there isn't a vessel available for washing with is there another option available for someone to eat bread? Or said another way if the only option for washing my hands is the sink but no cup available what's the proper procedure if any to be able to consume bread?
This is subject to a dispute among modern day poskim if the faucet has a din of a kli. Rav Elyashiv and Rav Ovadia Yosef both held it did have a din of a kli; thus, if no cup is available, one can turn the faucet on and off twice (or three if Sephardic) on each hand, since turning on the faucet is considered "koach gavra".
This is not to be relied upon practically: ask a rav personally.
One can dip one’s hands into a body of water that would otherwise be suitable as a mikvah. One may wash with other liquids, if necessary, though without a Bracha. (Shulchan Aruch 160:12)
While disposable cups are not ideal, one can use them if that’s all one has. Likewise, one may use a bottle, providing the water comes out in an uninterrupted flow (Mishna Berura 162:30).