[1]:http://pages.bangor.ac.uk/~mas010/topgpds.html
[2]:http://pages.bangor.ac.uk/~mas010/gpds-mv.jpg


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][1], and was in the 1988 (differently named) edition.

Edit: Here is an extract from the above paper

![mv][2]

which shows that there is some more information from the sequence than just the usual  exact sequence. This sequence applies to spaces as is shown in Section 4 of the above paper.  The point is that this detailed exactness is easier to extract in the groupoid model than directly in the topology.