No.

Choose a basis $V_1, V_2, V_3$ of Killing fields

Note that $$\int\limits_{\mathbb S^2} V_iu\cdot x_j\cdot d{\mathrm area} = -\int\limits_{\mathbb S^2} u\cdot V_ix_j\cdot d{\mathrm area}$$ Threfore is sufficient to find a fnction which is orthogonal to each of 9 functions $V_ix_j$.

The answer is "no".

Choose a basis $V_1, V_2, V_3$ of Killing fields.

Note that $$\int\limits_{\mathbb S^2} V_iu\cdot x_j\cdot d\,\mathrm{area} = -\int\limits_{\mathbb S^2} u\cdot V_ix_j\cdot d\,\mathrm{area}$$ Threfore you can take any $u$ which is orthogonal to each of 9 functions $s_{i,j}=V_ix_j$.