Let $M$ be an $n$-dimensional closed oriented connected manifold and suppose that $\bar{U},\bar{V}\subset M$ are $n$-dimensional manifolds with boundary so that $M=\bar{U}\cup \bar{V}$ and $\bar{U}\cap \bar{V}=\partial U=\partial V$.
If we compute Mayer-Vietoris sequence of $(\bar{U},\bar{V}, M)$ then the first non-trivial term looks like $$ 0\to \mathbb{Z}\overset{\delta}\to \mathbb{Z}^N \to \cdots $$ where $N$ is the number of components of $\partial U=\partial V$. I'm pretty sure that the map $\delta$ is given algebraically by $$1\mapsto (1, \ldots, 1)$$ where here $1$ represents the fundamental class of $M$ and $(1,0,\ldots, 0), \ldots, (0, \ldots, 0, 1)$ are the fundamental classes of the components of $\partial U$.
However, I don't work in topology and so am not completely confident of this. Is this correct and is there a reference for this sort of material?
Yes, if you think about $\bar U$ and $\bar V$ as about classes in $C_n(M)$, then $d(\bar U)=-d(\bar V)$ and $\bar U+\bar V$ equals the fundamental class of $M$. So you have $\delta :[M]\mapsto[\partial U]$, and fundamental class of $\partial U$ is the sum of fundamental classes of all components.
For more details i can advice just https://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence#Unreduced_version