Does there exist a contractible fiber bundle with fiber $G(\infty)$ and base $SU(\infty)$?

I'm not completely sure that I understand your notation, so this may be not what you want, but in case it is close enough, I'll have a go. The bit I'm assuming is that $G_m$ is the Grassmannian of $m$-places in $\mathbb{C}^{2m}$. That seems fairly safe, but my brain is refusing to check the homotopy types of everything involved at this hour.

I think that you can do this if you combine a construction of mine from The Co-Riemannian Structure of Smooth Loop Spaces with some bits from Loop Groups by Pressley and Segal, and then check a few details about how stuff holds together in the limit.

The bit that you need from Loop Groups is that the polynomial loop group, $\Omega_{\text{pol}} S U(n)$ acts on the finite restricted Grassmannian $Gr_0(H)$. Let me remind you what that space is: we start with the polarised Hilbert space, $H = L^2(S^1,\mathbb{C}^n)$, polarised as $H = H_+ \oplus H_-$ where $H_+$ are those functions with only positive Fourier coefficients and $H_-$ with strictly negative Fourier coefficients. From this, we define

$$ Gr_0(H) = \{W \subseteq H : \exists k : z^k H_+ \subseteq W \subseteq z^{-k} H_+\} $$

This is the union of $Gr(H_{-k,k})$ where $H_{-k,k} = z^{-k}H_+/z^kH_+$ so this (if I'm reading things aright) is the $G(\infty)$ of your question (Loop Groups, section 7.2).

The next bit that we need is that $\Omega_{\text{pol}} SU(n)$ acts on this space. This is from Theorem 8.3.2 and Proposition 8.3.3 in Loop Groups.

Putting these two together, if we have a principal $\Omega_{\text{pol}} SU(n)$-bundle over a space then we get a $G(\infty)$-fibre bundle over said space.

So now comes the bit from my work. In Section 3.2.3 of The co-Riemannian Structure of Smooth Loop Spaces, I construct a principal $\Omega_{\text{pol}} SU(n)$-bundle over $SU(n)$, with the property that under the obvious inclusion $\Omega_{\text{pol}} SU(n) \to \Omega SU(n)$ then this becomes the bundle coming from the usual path construction (so, although I don't need this in that paper, Bott periodicity implies that it is contractible).

So now we have a fibre bundle with fibre $G(\infty)$ over $SU(n)$. The last bit that you need is to show that under the inclusion $SU(n) \to SU(n+1)$ then these fibre bundles are compatible. The bit where this needs care is in the action of $\Omega_{\text{pol}} SU(n)$ on $Gr_0(H)$. But I think that if you include $L^2(S^1, \mathbb{C}^n)$ in to $L^2(S^1, \mathbb{C}^{n+1})$ at the same time, then you should get a diagram that works. The resulting Grassmannian will be $\bigcup Gr_0(L^2(S^1,\mathbb{C}^n))$ but that will still be $G(\infty)$ (assuming that I've understood the question correctly).

As I said, there's a few ifs and buts here: if I understood the question correctly, and if everything holds together in the limit (but I'm pretty sure that the second "if" is fine), but obviously as I'm less sure about the first if I haven't checked all the details.