I am taking my definitions from Wikipedia.
On the Differentiable manifold Wikipedia page, a chart on a differentiable manifold $M$ is defined as a pair $(U,\varphi)$ of an open subset $U\subseteq M$ and a homeomorphism $\varphi$ from $U$ to an open subset of $\mathbb{R}^n$.
On the Tangent space Wikipedia page, the tangent space $T_xM$ of the manifold $M$ at $x$ is constructed as the quotient set of differentiable curves on $M$ through $x$, where two curves are equivalent if they have the same derivative in $x$. To obtain the vector space structure on $T_xM$, they say to pick a chart $\varphi:U\to\mathbb{R}^n$ and to define a map $d\varphi_x:T_xM\to\mathbb{R}^n$ by setting $d\varphi_x([\gamma]) = (\varphi\circ\gamma)'(0)$ (where $\gamma$ is a differentiable curve on $M$ with $\gamma(0) = x$).
It is then said that this map $d\varphi_x$ is bijective. The fact that it is injective is obvious to me, as it follows immediately from the definition of the equivalence relation on the set of differentiable curves through $x$. And as for surjectiveness, given a vector $\vec{v}\in\mathbb{R}^n$, one can pick the curve $t\mapsto \varphi^{-1}(\varphi(x) + \vec{v}t)$.
However, it seems to me that this surjectiveness argument requires the image of the chart to be the entirety of $\mathbb{R}^n$, since we can otherwise not guarantee that the image of the curve lies within $U$. So this has me confused, since the first definition of a chart says that the image only needs to be an open subset of $\mathbb{R}^n$.
How do you prove that $d\varphi_x$ is surjective if $\varphi$ only maps to an open subset of $\mathbb{R}^n$, and not the entire space? Or is a different definition preferable in that case?
No, $\phi$ does not have to map onto $\Bbb{R}^n$. Just to be clear, the definition of $T_xM$ is that it is a quotient of the set of all smooth curves $\gamma:I\to M$ where $I$ is an open interval containing the origin and such that $\gamma(0)=x$. Note that it is not required $I=\Bbb{R}$.
The mapping $t\mapsto \phi^{-1}(\phi(x)+tv)$ needs only to be defined on a small open interval containing the origin, as required by the definition. This curve is well defined for the following reason: say $\phi:U\to \phi[U]\subset\Bbb{R}^n$ is the chart map. Given $x\in U$, we have $\phi(x)\in \phi[U]$, so because $\phi[U]$ is open in $\Bbb{R}^n$, there is an $\epsilon>0$ such that the open ball $B_{\epsilon,\phi(x)}$ of radius $\epsilon$ and center $\phi(x)$ is contained in $\phi[U]$. Thus, the curve $\gamma: \left(-\frac{\epsilon}{\|v\|},\frac{\epsilon}{\|v\|}\right)\to M$ given by $\gamma(t)=\phi^{-1}(\phi(x)+tv)$ is well-defined (if $v=0$, we interpret the domain of $\gamma$ as $\Bbb{R}$, in which case it is simply the constant map $\gamma(t)=x$).