The best know classical Indian school which enforces a concept of rationality is the school of Nyāya.
Its main method is critical question and argumentation on the basis of logic, the Sanskrit word is ānvikṣikī. The canonical source is the nyāya sūtra. Different sections of the nyāya sūtra deal with
Knowledge sources, Doubt and the philosophical Method, In Defense of
the Real, Self, Substance and Causation, God, Word and Object, The
Right and the God, Debate.
This list is taken from the book The Nyāya sūtra. Selections with Early Commentaries by Matthew Dasti and Stephen Phillips. This book gives an introduction on the level of an undergraduate course.
A monography written on graduate and research level is Philosophy in Classical India. The proper work of reason by Jonardon Ganeri.
A good textbook with a broad scope is An Introduction to Indian Philosophy by Roy Perrett. The book follows the work of Ganeri.
I do not see important differences between the epistemology developed in the Nyāya school and e.g. the main chapters of the Metaphysics by Aristotle. Both works follow consequently a rational methodology.
But I do not find in Indian texts on epistemology any analogoue to analytic philosophy or to Popper's fallibilism.