Consider it from the perspective of the conservation of energy:
2H2 + O2 = 2H2O + energy
The energy is on the right because the reaction is exothermic.
The opposite reaction must be endothermic and must have the same amount of energy on the left:
2H2O + energy = 2H2 + O2
Otherwise, energy isn't conserved.
Because nothing is 100% efficient, there's also energy lost to inefficiencies. So the answer to your question is that yes, you need more energy to make the second reaction happen than is stored in the chemical bond.
Comments
Consider it from the perspective of the conservation of energy:
2H2 + O2 = 2H2O + energy
The energy is on the right because the reaction is exothermic.
The opposite reaction must be endothermic and must have the same amount of energy on the left:
2H2O + energy = 2H2 + O2
Otherwise, energy isn't conserved.
Because nothing is 100% efficient, there's also energy lost to inefficiencies. So the answer to your question is that yes, you need more energy to make the second reaction happen than is stored in the chemical bond.