(Edited after taking into account Tom Goodwillie's answer.)

Let $E \rightarrow X$ be an orientable vector bundle.

In [this MO answer][1] it is explained how to obtain a representative of the Thom class (call this a Thom form) which pulls back to the natural Euler-form given by Chern-Weil theory. This form depends on the choice of: a metric and a compatible connection on $E$, and a cutoff function $\rho:[0,\infty)\rightarrow [0,\infty)$ which is equal to $−1$ near $0$ and equal to $0$ on $[1,\infty)$.

If we are given a Thom form for $i^*E \rightarrow \partial X$ constructed via the above method, one can extend the metric and connection to $E \rightarrow X$ using a partition of unity argument, and use the above method to obtain a Thom form for $E \rightarrow X$ extending the Thom form on the boundary. However, it is not clear to me whether every choice of Thom form can be constructed using this method.

As answered by Tom Goodwillie in the comments, if $i: \partial X \hookrightarrow X$ is a smooth manifold with boundary and $E \rightarrow X$ an orientable vector bundle then any Thom form for $i^*E \rightarrow \partial X$ can be extended to a Thom form for $E \rightarrow X$. But, this extension might not arise from a metric and compatible connection.

If this general question proves too hard, I am trying to use it to obtain an answer to the following more specific situation: suppose that $i^*E = E_1 \oplus E_2 \rightarrow \partial X$, then one can choose metrics and compatible connections $g_i$ and $\nabla_i$ for $E_i$, and a function $\rho$ as before, and use this to obtain Thom forms $\tau_i \in \Omega^{rk(E_i)}_{cv}(E_i)$, then $\tau_1 \wedge \tau_2$ is a Thom form for $i^*E \rightarrow \partial X$. Are there choices of metric and connection which give rise to such an extension? One might hope to prove this directly from the formula for the Thom form, but it is not clear to me what happens in the construction of the global angular form of a direct sum.


  [1]: https://mathoverflow.net/questions/117970/is-there-a-natural-form-representing-the-thom-class-of-a-vector-bundle-which-wh