Let $S$ be a smooth quasi-projective scheme over $\mathbb{Z}$, and $A$ an abelian scheme over an open subset $U \subset S$. Suppose that $S\backslash U$ has codimension at least $2$ and that for every point $s \in S$, the ramification index at the stalk $\mathcal{O}_{S,s}$ is $1$, so that the hypotheses for the Faltings-Chai and Vasiu-Zink criteria hold at each stalk. Then

1. Can $A$ be extended to an abelian scheme over $S$?
2. If $A$ comes equipped with a principal polarization $\lambda$, and $A$ extends to an abelian scheme $B$ over $S$, can we extend $\lambda$ to a principal polarization on $B$?

Take a generic point $\eta$ of an irreducible component of $S \backslash U$. The stalk $\mathcal{O}_{S,\eta}$ has a map $f:\textrm{Spec}(\mathcal{O}_{S,\eta}) \to S$ and we have an abelian scheme $f^*A$ defined outside the maximal point of $\mathcal{O}_{S,\eta}$. We are in a position to apply either Faltings-Chai or Vasiu-Zink to extend $f^*A$ uniquely to an abelian scheme $A'_{\eta}$ over the stalk. We can then spread this out to an abelian scheme $A'_{U_{\eta}}$ over some open $U_{\eta}$ containing $\eta$.

For any other point $\zeta$ so that $\eta \in \overline{\zeta}$, we know $\zeta \in U \cap U_{\eta}$ and if $g: \textrm{Spec}(\mathcal{O}_{S,\zeta}) \to S$ is the natural map then $g^*A \cong g^*A'_{U_{\eta}}$ (abusing notation slightly). We can spread this out to an isomorphism of abelian schemes $g_{\zeta} : A |_{W_{\zeta}} \to A'_{U_{\eta}}|_{W_{\zeta}}$ over some open $W_{\zeta}$ around $\zeta$.

As long as the cocycle conditions are satisfied, we would be able to glue $A$ and $A_{U_{\eta}}'$, after potentially shrinking $U_{\eta}$, to an abelian scheme over a strictly larger open $U \cup U_{\eta}$. The logic behind the previous paragraph ensures that $\eta$ remains in $U_{\eta}$ after shrinking. Now a Zorn's lemma type argument allows us to conclude (1).

The cocycle conditions hold at least generically (after taking generic fibre above $S$), so by rigidity of abelian schemes, I suspect that they hold, and the question (1) is indeed true.

As for (2), the line bundle $L$ on $A$ which induces the principal polarization can be extended over $B$ to some $\overline{L}$, and we obtain a morphism $\phi_{\overline{L}}:B \to B^{\vee}$. I suspect that relative ampleness can be verified pointwise on $S$, and supposing that $\overline{L}$ is a polarization, it must be principal since it is so generically.