We define two algebra automorphisms $T_0$ and $T_1$ of the q-Onsager algebra Bc, which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a PBW basis for Bc. We show that the root vectors satisfy q-analogs of Onsager's original commutation relations. The paper is much inspired by I. Damiani's construction and investigation of root vectors for the quantized enveloping algebra of $\widehat{sl_2}$.