He was born in St Peter-in-the-East, Oxford, the first son of the Rev Dr James Bandinel, of Netherbury, and Margaret Dumaresq. Educated at Reading under Richard Valpy and then at Winchester College, Bandinel entered New College, Oxford, in 1800 (BA 1805, MA 1807, BD and DD 1823) and was a Fellow there until 1813. He was ordained as a priest in the Church of England in 1805.
During Admiral Sir James Saumarez's Baltic campaign of 1808, Bandinel served a short while as chaplain on board HMS Victory; but, returning, he settled in Oxford and rose within the University's ranks. From 1810 he was Sub-Librarian of the Bodleian Library under his godfather John Price, and rose to become Bodley's Librarian in 1813 upon Price's death. It was a position he held until his own death in 1861. Bandinel was Dean of New College and Proctor of the University in 1814, and a Delegate of the University Press from 1813.
The Bodleian's collections increased greatly under his direction and his knowledge of literary circles was rarely seconded. His patience with both ill-informed library visitors and colleagues would often run thin, many a guest falling victim to his short temper, but it is said that his courtesy was guaranteed to anyone of note who wished to consult him.
Bandinel married, in 1813, Mary Phillips, daughter of John Phillips of Culham, Oxfordshire, and died without issue in 1861 at his home, 31 Beaumont Street, Oxford.