I somewhat can wrap my head around virtual particles in terms of quantum field theory, (as much as possible), and I'm just curious how string theorists would describe them.
In Quantum Field Theory virtual particles are not particles, they are just disturbances in the field that is not a particle itself. So you have an effect that looks like its coming from particle but there is no particle. We measure the effect and use that for calculations.
Now in string theory, we just have strings that are vibrating at different frequencies. We treat them as "real" strings. Then there are virtual strings, we can only measure(for lack of better word) the effects but they are not the "real" strings that we defined but are "virtual" strings but we use the effect in our calculations.