Hints:

1. The condition for cancellation of tadpoles can be formulated as vanishing of 1-point functions
$$ \langle \phi(x)\rangle_{J=0}~=~0\qquad\text{and}\qquad\langle \chi(x)\rangle_{J=0}~=~0,$$
cf. e.g. my Phys.SE answer [here](https://physics.stackexchange.com/a/111381/2451). 

2. The first condition is automatic satisfied due to a $U(1)$-symmetry of the complex field $\phi$. 

3. The second condition can be made to hold by adding a 1-vertex interaction term $Y\chi$ to the Lagrangian density, cf. Ref. 1.

References:

1. M. Srednicki, _QFT, 2007;_ chapter 9. A prepublication draft PDF file is available [here](http://web.physics.ucsb.edu/~mark/qft.html).