I would say, it depends on what you mean by: "Is the local inverse Galois problem is solved?"

One way to formulate the local inverse Galois problem is, for any given a finite group $G$ can we decide whether $G$ is the Galois group of a local field. Or more algorithmically:

Does there exist an algorithm that has as input a finite group G and a
local field K and outputs whether or not the case that the group G is
the Galois group of an extension L/K.

If you formulate it this way the answer is yes. And the reason for this maybe not as nice as one would like.

The reason why the answer is yes is as follows: For a given a group $G$ there are up to isomorphism only finitely many extensions of $L/K$ of degree $\#G$. Indeed, when weighted correctly we even know explicit formula for the number of extensions and there are algorithms that explicitly enumerate these extensions. So the only thing one has to do is compute the Galois group of every extension $L/K$ of degree $|G|$, and see whether it is isomorphic to $G$.

This algorithm can even be made "independent of the characteristic. I.e. for a given group $G$ on can explicitly decide for which characteristics $p$ the group $G$ can be a Galois group of $\mathbb Q_p$. The reason is that when $L/\mathbb Q_p$ is tamely ramified we know very well which groups occur (see Iwasawa's result in Jeremy's answer). So using this it is reduced to doing a finite computation as above for all the primes $p | \#G$ where there can be wild ramification.

So it all depends on how you formulate the question "Is the local inverse Galois problem solved?". Cause it all depends on what way of characterizing the groups that can occur as a local Galois group you are fine with.

**Edit**
I just found out about the existence of the paper The Inverse Galois Problem for p-adic fields
by David Roe. His results are only for when the residue characteristic is not $2$ since he uses the explicit description of the absolute Galois group of Jannsen and Wingberg mentioned by Jeremy. In that paper he gives an algorithm to actually count the number of extensions of K with galois group G, which is more efficient than the enumeration I described above. Additionally he gives a set of necessary and a distinct set of sufficient conditions for a group G to be realizable as a Galois group of over K, which might be of independent interest.

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