Is $S^1\vee\ldots\vee S^1$ a deformation retract of $\mathbb R^2-\{x^1,\ldots,x^n\}$?

For two points it is very easy. Let $x_0=(0,0), x_1 = (2,0)$. Identify $S^1 \vee S^1$ with $C_0 \cup C_1$, where $C_i$ is the circle with center $x_i$ and radius $1$. Let $D_i$ be the closed disk with center $x_i$ and radius $1$. Then $D_i \setminus \{x_i\}$ strongly deformation retracts to $C_i$ (radial retraction to the boundary). The "exterior" $E = \mathbb R^2 \setminus \operatorname{int}(D_0 \cup D_1)$ strongly deformation retracts to $L = C_0 \cup C_1 \cup (-\infty,-1] \times \{0\} \cup [3,\infty) \times \{0\}$ (shift each point of $E$ vertically in direction of the $x$-axis until it reaches $L$). But $C_0 \cup C_1$ is clearly a strong deformation retract of $L$.

For more than two points it is technically more complicated, but can be done.