Is it possible to describe the Local F-theory models which seems to be relevant for particle physics phenomenology using Gauged Linear Sigma Models (GLSM), this includes describing the decoupling limit using the parameters of the GLSM? Actually, I am not sure if my question makes sense. Another related question is: Can elliptically fibered Calabi-Yau 4-folds be described as the IR limit of GLSM while also producing the global constraint of F-Theory compactifications, and also producing the Weierstrass form of the elliptic fibre. Thanks in advance.
concerning the first question, I don't think so. If I understand you well, you want to use GLSM's for a world sheet description of the vacuum. But a world sheet description of any string theory is only possible if the coupling constant in almost all of space is weak because the world sheet is the history of a (weakly coupled) string - so strings must remain the lightest objects etc.
However, the very point of F-theory - including the loci of the branes in the local models - is that the complexified coupling $\tau$ living in the fundamental domain may have very different values across the space, and usually undergoes nontrivial monodromies - which show that the strong coupling is identified with the weak coupling. So the world sheet of the fundamental string is exchanged with the world volume of the D-strings, and so on. That's why people have to use spacetime methods for F-theory.
I don't know the answer to the elliptically fibered question you posed.