Fundamental group of a wedge sum, in general (e.g. when van Kampen does not apply)

Under certain conditions they do actually mean the existence of neighborhoods, containing the common point $x_0$ of the wedge, which do retract on $x_0$.

With this condition we are able to choose suitable open path-connected sets $A_i$ to compute the desired fundamental group.

For instance if you take the $n$ wedge of circles $S^1$ at $x_0$ you can compute its fundamental group via Van-Kampen. What is important here is that we have open neighborhoods $N_i$ containing $x_0$ for each circle $S^1$ of the wedge (just imagine a little open arc on $S^1$ which contains the center $x_0$ of the wedge) which do actually retract on $x_0$ (just shrink from both sides until you reach the center $x_0$ of the wedge).

Now you can indeed define $A_i$ to be the wedge of all $N_j$ for $j \neq i$ and wedge $S^1$ i.e. $$A_i=\left(\bigvee_{i \neq j}N_j\right) \vee S^1.$$ If you now take two different $A_i$'s, i.e. $A_i$ and $A_j$ and look at the intersection $A_i \cap A_j$ you'll see that it's simply the wedge of all $n$ neighboorhoods $N_i$ of $x_0$ and since all of them retract to $x_0$ we get $$\pi_1\left( \bigvee_{i=1}^nN_i\right) \cong \pi_1(\{x_0\})=0.$$

By a similiar argument using your retracts you can compute $$\pi_1(A_i) \cong \pi_1(S^1) \cong \mathbb Z.$$

Now by applying Van Kampen you get that the fundamental group of your wedge sum is free abelian of rank $n$.