Does feeling shy about supplicating to Allah in front of people, and refraining from raising one's hands during supplication for fear of being seen by others, constitute a weakness in faith or a reason for the supplication not to be answered?

If your shyness in raising your hands during supplication is due to fear of showing off (riyaa'), do not indulge it, because abandoning a good deed out of fear of showing off is blameworthy. However, if it is shyness because it is not common in your society, then this is blameworthy shyness, and you should abandon it and continue supplicating and raising your hands with a good intention and a desire to spread good. And if you do not raise your hands, then supplication by merely moving the tongue is permissible, because raising the hands is a Sunnah (definition), not an obligation.