Does string theory state as vibrational entropy increases, mass increases?
I think that your question is about string thermodynamics in the microcanonical ensemble. The entropy obtained by calculating the partition function of a relativistic quantum string is just the Boltzmann constant multiplied by the number of vibrational microstates, calculated by using the Hardy-Ramanujan asymptotic expansion. The entropy energy relation in relativistic open and closed strings is worked in detail in chapter 16 of Barton Zwiebach book on string theory. It turns out that entropy is directly proportional to energy. This means that increasing vibrational energy (and also mass) the entropy increases linearly. So mass increased with vibrational entropy. The curious thing is that the derivative of energy with respect to entropy is the temperature of the strings thermodynamic system. This temperature is then constant; no matter how large is the energy! So the answer to your question, according to this reference, is yes, in the high energy regime, mass increases with entropy. I hope this is the answer you were looking for.