Does string theory have a notion of vacuum? If yes, what is known about it?
As I recall from Susskind's course, there is no actual vacuum in string theory. There are some pieces of information, which can be helpful, like terminology developed for 2 decades. Please, note the dates.
String theory is believed to have a huge number of vacua — the so-called string theory landscape.
Terminology starting from almost nothing:
Developed to more specific ideas:
Can't find any other notions dated after 2007. Hope this was helpful.
In the string theory literature, the term "vacuum" is usually synonymous with "perturbative string background", i.e. a target space of a 2d CFT with the right central charge. This target space comes equipped with a metric satisfying the Einstein equations, as well as a host of other background fields, dilaton, B-field, and whatnot, all satisfying the equations implied by conformal invariance on the worldsheet.
Given any such background, you can use the machinery of perturbative string theory to add excitations. A standard (but not terribly rigorous) argument indicates that adding these new excitations is equivalent to deforming the background fields. Which means that the new state with excitations is also a string background. So any string theory state is a string background, and hence in some sense, a vacuum state.
It seems reasonable to think that any string background is a coherent state, built up by adding arbitrarily many strings to a state with no strings at all. This hypothetical no-string state would be a better analogy to the usual sort of QFT vacuum. But it's never been defined. We don't understand what string theory is well enough to say if this true vacuum exists, or if all we can do is dial around through non-trivial backgrounds.
Similar comments apply to other corners of string theory, like matrix models and AdS/CFT.