In general the answer is negative as it follows by the counterexample provided by Bolibruh (or Bolibruch, see wikipedia: Hilbert 21).
Nowdays generalizations and refinements of this problem are called the Riemann-Hilbert problem. For example techniques of finding the corresponding ODE when it is possible, the closely related Riemann-Birkhoff factorization (realization of a holomorphic matrix function of a circle as a product of a matrix holomorphic on a neighborhood of closed disk and a function of a matrix holomorphic on a neighborhood of an exterior of the disk including infinity and the circle itself). The correspondence between differential equations and monodromies can in fact be established and is true in general if we understand the data much more generally, using sheaf theory. Namely, that is a rough meaning of so-called Riemann-Hilbert correspondence (wikipedia) in sheaf theory on complex manifolds.
In integrable systems theory (and study of special functions) one often says Riemann-Hilbert method.
While the short wikipedia article on 21st Hilbert problem is discussing in nice generality, the page titled Riemann-Hilbert problem, which has more formulas, is in fact explaining only the variants of the Riemann-Birkhoff factorization aspect; it has however a nice list of application areas.
N. Katz, An overview of Deligne’s work on Hilbert’s twenty-first problem, Proc. of Symp. in Pure Math. 28, 537-557 (1976).
O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Univ. Press 2003.
G. D. Birkhoff, The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Amer. Acad. Arts and Sci. 49 (1913), 531-568.
D. V. Anosov, A. A. Bolibruch, The Riemann-Hilbert problem, Aspects of Math. E22. Friedr. Vieweg & Sohn, Braunschweig, 1994. x+190 pp.
Alexander R. Its, The Riemann-Hilbert problem and integrable systems, Notices Amer. Math. Soc. 50 (2003), no. 11, 1389–1400, (survey) pdf
E. Corel, E. Compoint, Stable Flags and the Riemann-Hilbert Problem, arxiv/1003.5021
R. R. Gontsov, V. A. Poberezhnyi, Various versions of the Riemann–Hilbert problem for linear differential equations Russian Mathematical Surveys 63:4 (2008), 603–639 (English bibliography free here: pdf)
A.A. Bolibrukh, The Riemann–Hilbert problem, Russian Math. Surveys 45:2 (1990), 1–58; Rus. original А. А. Болибрух, “Проблема Римана–Гильберта”, УМН 45:2 (1990), 3–47 (pdf)
The AGT correspondence is treated with the help of a Riemann-Hilbert problem in