The charge of a particle is what determines the force that it experiences under the influence of an Electric field. The acceleration of the particle also involves the mass of that particle.
Consider constant electric field E.
F = qE
Where
F-force on the particle
The particle’s q-charge
Because electron and proton have the same charge (+e) and-e, the force acting upon them is the same (E is constant).
Let a be acceleration, then
a = F/m
Where m is its mass.
Because proton’s mass is much larger than that of electrons, the acceleration of proton will not be as strong as an electron’s.
P.S. – Because of the different sign of charge, acceleration of electron and proton will occur in an opposite direction.