Your metric has to be geodesic, in particular 
$$d(x,y)=\min \{\,1,|x-y|\,\}$$
is not Finsler in your sense.

Now let $d$ be a geodesic metric on $\mathbb{R}^n$.

Suppose $n=2$.
In ["Two counterexamples..." by Burago, Ivanov, and Shoenthal][1], it was conjectured that a neighborhood of any point in $(\mathbb{R}^2,d)$ admits a Lipschitz embedding into the Euclidean plane.

Suppose $d$ is a Finsler metric in your sense.
Since $\phi$ is continuous, the natural map $(\mathbb{R}^n,d)\to \mathbb{R}^n$ is a locally lipschitz homeomorphism.
So this conjecture is closely related to your question.

Now suppose $n=3$.
The same paper provides an example of a metric $d$ on $\mathbb{R}^3$ that (locally) does not admit a Lipschitz embedding into $\mathbb{R}^3$.
(The construction is interesting --- it is worth reading.)

[1]: https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=102&option_lang=eng