Fundamental group of a topological pullback

There is a ``Mayer--Vietoris" sequence $$\cdots \to \pi_2(Z, z) \to \pi_1(E, e) \to \pi_1(X, x) \times \pi_1(Y, y) \to \pi_1(Z,z) \to \pi_0(E) \to \cdots$$ that can be developed by fitting together the 4 long exact sequences of homotopy groups obtained from the 4 maps in your diagram.

With $X \simeq Y \simeq *$ and $Z=S^2$, $E \simeq \Omega S^2$ has fundamental group $\mathbb{Z}$, whereas $X$, $Y$ and $Z$ are simply-connected, so you can't express $\pi_1(E)$ solely in terms of the fundamental groups of the other spaces.

Expressed in terms of the homotopy pullback $N(f,g)$ of a pair of based maps $f\colon X\longrightarrow A$ and $g\colon Y\longrightarrow A$, the long exact sequence is Corollary 2.2.3 of May and Ponto ``More concise algebraic topology''. The result of which it is a corollary, Proposition 2.2.2, describes the pointed set of based maps $[Z,N(f,g)]$ for any based space $Z$. The dual result for homotopy pushouts is Proposition 2.1.2. The corollary is used heavily in the study of fracture theorems for localization and completion later on in the book.

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.

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Edit: Here is an extract from the above paper

(source)

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.