A few bits and pieces of Fredholm theory in the locally convex (and, in particular, Fréchet) setting are discussed in the literature. The most comprehensive treatment I could find were two old articles by Schaefer:
There he discusses how the basic elements of Fredholm theory like invertibility modulo compact operators, existence of a parametrix and the Fredholm alternative generalize from Banach to locally convex spaces.
However, most applications of Fredholm theory use families of operators and things like the invariance of the index under small deformations etc. These results do not directly generalize to locally spaces since the set of invertible operators fails to be open, leading to many pathological (?) counterexamples. In order to discuss families of Fredholm operators and extend the important results about them to the locally convex setting, in my Phd Thesis, I introduced the notation of a "uniformly regular" family of operators. Roughly speaking, these are families of operators that posses a family of (generalized) inverses. This allows to extend most classical results to this setting. For example, if you have a uniformly regular family of operators and one of them happens to be Fredholm, then all of them are Fredholm with the same index. Finally, a classical result about elliptic operators (Theorem II.3.3.3 in Hamilton, R. S., The inverse function theorem of Nash and Moser) can be used to show that families of elliptic operators are uniformly regular so that these results apply in particular to elliptic operators.
A streamlined (but perhaps less pedagogical) exposition of uniformly regularity and Fredholm theory can be found in the article