The massive states due to vibrations on the string have masses given as integer multiples of the string scale, which you should think of as a very large scale, near the Planck scale. (At least, nearer to the Planck scale than it is to the TeV scale).
The particles in the Standard Model would actually correspond to massless particles, in "naive" (free) string theory. Therefore some mechanism other than string vibrations is needed to give Standard Model particles mass, even in string theory. For quarks, leptons, and gauge bosons the Standard Model, this is the Higgs mechanism, which arises due to interactions between the particles and a background scalar field.
Typically the way you would think about this in practice is to say that at low energies compared to the string scale, string theory can be well described as an effective field theory consisting of the "massless" states of the string (that is, states with masses much less than the string scale). In string theory, scalar fields often arise as effective low energy degrees of freedom describing the ways the extra dimensions of string theory are compactified. From this perspective, you can think of the Standard Model as an effective field theory that in principle could be derived from string theory in an appropriate limit. In string theory, scalar fields often arise as effective low energy degrees of freedom describing the ways the extra dimensions of string theory are compactified. Although this comes with a major caveat: no one has shown that our Standard Model actually can be derived as a low energy limit of string theory in some vacuum state.