Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$ a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section such that $ s_2 : s_1^{-1}(0) \rightarrow V_2$ is a smooth section, transverse to the zero set. Let $p\in M$ be a point such that $$ s_1(p)=0, \qquad s_2(p) =0. $$ Does there always exist a solution $p(t_1, t_2)$ near $p$ such that $$ s_1(p(t_1,t_2)) =t_1 \qquad s_2(p(t_1,t_2)) = t_2 $$ if $t_1$ and $t_2$ are sufficiently small? To make sense of sufficiently small we can just chose some metric on $V_1$ and $V_2$.

Note that if I said that the section $p \rightarrow s_1(p) \oplus s_2(p)$ is smooth and transverse to the zero set, then the answer is obviously yes (by implicit function theorem). It seems that one should be able to imitate the proof of implicit function theorem (ie contraction mapping principal) to justify what I am asking. But I am not sure of this.