(0) I do not know what is contained in a thesis which is not published and is not even available online (12 years after the defense).
(1) By Proposition 8.1.4 in the 1000-page FOOO book, a relatively spin structure on a (necessarily orientable) Lagrangian submanifold $L$ of a symplectic manifold $M$ determines an orientation on the moduli spaces of $J$-holomorphic disks $(D^2,S^1)\longrightarrow (M,L)$. The same reasoning applies to a family of real Cauchy-Riemann operators induced (as in Remark 1.3 of 1207.5471) by a bundle pair $(E,F)\longrightarrow(M,L)$, where $L$ is any submanifold of any manifold $M$. Proposition 8.1.7 gives an example of a non-orientable family of real Cauchy-Riemann operators, crediting it to Vin de Silva, and including a proof. I do not believe this book contains other, substantially different, statements on orientability in open GW-theory. Thus, this book discusses orienting moduli spaces of disks in some cases, but says fairly little about their orientability in general.
(2) By Theorem 1.1 in 0606429, a relatively Pin structure on a non-orientable Langrangian induces an isomorphism between the orientation line bundle of the moduli space of open $J$-holomorphic maps from Riemann surfaces with a fixed complex structure and a product of pull-backs of the orientation line bundle of the Lagrangian by evaluation maps. I do not believe this paper contains other, substantially different, statements on orientability in open GW-theory. Thus, this paper contains a number of results on both orienting and orientability of moduli spaces.
(3) Lemma 11.7 in Seidel's book does what Tim says in (1). Unfortunately, it requires more than a quick look to understand and see that it implies Proposition 8.1.4 in FOOO and the disk case of Theorem 1.1 in 0606429.
(4) Theorem 1.1 in 1207.5471 describes the holonomy of the orientation bundle of a family of real Cauchy-Riemann operators over bordered Riemann with varying complex structures. Its statement is absolutely clear from looking at the preceding half a page, at the beginning of the introduction. In particular, it is almost immediately clear that this theorem implies Proposition 8.1.4 in FOOO and the full statement of Theorem 1.1 in 0606429. The proof, contained in Section 3, is beautifully simple and uses no K-theory or even homotopy exact sequences.
(5) In the case of anti-symplectic involutions, one often wants to orient moduli spaces of real maps, not their halves, even if they are halvable. In the case of maps from $S^2$ with the standard conjugation, if the corresponding moduli space of disk maps is orientable, the space of real maps is orientable if the flip map on the disk space is orientation-preserving. So, this becomes a problem about computing the sign of this flip map. Results on this are Proposition 5.1 in 0606429 and Theorems 1.1 and 1.3 in 0912.2646. If the involution on $S^2$ has no fixed points, the orientation problem has nothing to do with any Lagrangians. Results on orientability in this case are Theorem 1.3 and Example 2.5 in 1205.1809 and Theorem 1.1 and Corollary 1.8 in 1301.1074.
(6) 1301.1074 says less about the Lagrangian case than 1207.5471. The point of 1301.1074 is to study the orientability problem for $J$-holomorphic maps that commute with involutions on the domain and the target. These maps need not be halvable to a map from a bordered Riemann surface with Lagrangian boundary conditions. An application is Corollary 1.8.