Have a look at
(R. BROWN, P.R. HEATH and H. KAMPS), ``Groupoids and the Mayer-Vietoris sequence'', J. Pure Appl. Alg. 30 (1983) 109-129.
A Mayer-Vietoris sequence for a pullback of a covering mao also appears in Section 10.7 of Topology and Groupoids, and was in the 1988 (differently named) edition.