I've gone through the constructions to change add many subsets to different cardinals, and know that Easton's theorem says that the power function can consistently be anything not inconsistent with basic cardinal arithmetic or Konig's lemma. This says that, for example $2^{\aleph_0}$ cannot be forced to be $\aleph_{\omega}$.
My question is then, what exactly goes wrong in the construction if you try to obviously modify Cohen's construction to make this happen? What's the problem with forcing over a ctm $M$ with $\mathbb{P}$, the set of finite partial functions $\aleph_\omega^M \times \omega\to \{0,1\}$, and just adding in $\aleph_{\omega}$-many reals that way?
You will add $\aleph_\omega$ new reals, but that only gives a lower bound for the continuum (the Cohen reals you add will not be the only new reals in your model!). To figure out exactly what the continuum becomes, you also need to get an upper bound. There is some combinatorics you can do with counting names for reals that gives an upper bound of $(\kappa^{\aleph_0})^M$ reals in the extension, if you add $\kappa$ Cohen reals. The usual way this is applied is when you know that $\kappa^{\aleph_0}=\kappa$ in the base model, so this gives you exactly $\kappa$ reals in your extension. But when $\kappa=\aleph_\omega$, it's impossible to have $\kappa^{\aleph_0}=\kappa$, so you can't use this to get a model with exactly $\aleph_\omega$ reals.