Distributions valued in Fréchet spaces A distribution on $(a,b)$ with values in a Banach space $X$ is a linear functional $u : \mathcal{D}(a,b)\to X$ if: 
For every compact set $K\subset (a,b)$, there is a real  number $C\geq 0$ and a nonnegative integer $N$ such that:
$$\|\langle u, \phi \rangle\| \leq C \sum_{\lvert \alpha\rvert \leq N } \sup \lvert \partial^\alpha \phi\rvert  $$ 
for all $\phi \in \mathcal{D}(a,b)$ with $\mathrm{supp}(\phi) \subset K$, where $\| \cdot\|$ is the norm on $X$.

Assume instead that $\mathcal{F}$ is a Fréchet space with semi-norms $(p_1, p_2, \ldots)$: what's the definition of a distribution on $(a,b)$ with values in $\mathcal{F}$?

 A: I'll share what I've found. The next results are taken from Gerd Grubb's book: Distributions and operators (see Appendix B and Chapter 2). 
We recall that:


*

*$D(a,b)$ is the inductive limit of $C^{\infty}_{K_j}(a,b)$, where $K_j$ is an increasing sequence of compact subsets of $(a,b)$ and $C^{\infty}_{K_j}$ is a Fréchet space with the usual topology inherited from $C^{\infty}(a,b)$.

*Let $Y$ be a locally convex topological vector space. A linear map $T: D(a,b) \to Y$ is continuous if and only if $T:C^{\infty}_{K_j}(a,b) \to Y$ is continuous for every $j \in N$.

*A family of seminorms $\mathcal{P}$ has the max-property if: 
$$ \forall \, p_1, p_2 \in \mathcal{P}\quad \exists p\in \mathcal{P}, c>0 : p\geq c \max\{p_1,p_2\} .$$

*Let $X$ and $Y$ be Fréchet spaces, with corresponding family of seminorms $\mathcal{P}$ and $\mathcal{Q}$. Then a lineare operator $T$ from $X$ to $Y$ is continuous if and only if: For each $q \in \mathcal{Q}$ there exists a $p \in \mathcal{P}$ and a constant $c>0$ so that: for all $x \in X$
$$\lvert q(Tx)\rvert \leq c p(x).$$
Using the above facts, we have:
Let $\mathcal{F}$ be a Fréchet space with the topology induced by the family of seminorms $\mathcal{P}$ having the max-property, then a a linear operator $T: D(a,b) \to \mathcal{F}$ is a distribution if and only if: 
For any $p \in \mathcal{P}$ and for any $j \in  \mathbb{N}$, there exist $N \in  \mathbb{N}_0$ and $C>0$ such that
$$\lvert p( T \varphi)\rvert \leq C \sup\{\lvert\partial^\alpha \varphi(x)\rvert \;| x \in K_j, \lvert\alpha\rvert \leq N \}$$
for any $\varphi \in C^{\infty}_{K_j}(a,b)$.
Standard reasonings then implies that this is equivalent to the sequential continuity as suggested in the comments.
A: Fact 1. Let $V$ and $W$ be locally convex vector spaces with families of seminorms $(r_k)_{k\in L}$ and $(q_\beta)_{\beta\in B}$, respectively. A linear map $T:V\to W$ is continuous iff for every $\beta\in B$, there exist $C \geq 0$ and $k_1,...,k_n\in L$ such that 
$$q_\beta(Tv)\leq C\sum_{j=1}^n r_{k_j}(v),\qquad \forall\ v\in V.$$
Fact 2 (see p. 14). Let $W$ be a locally convex vector space. A linear map $T:\mathcal{D}(a,b)\to W$ is continuous iff $T:C_{K_j}^\infty(a,b)\to W$ is continuous for all $j\in\mathbb N$, where the (locally convex) topology of $C_{K_j}^\infty(a,b):=\{\phi\in C^\infty(a,b)\mid \text{supp}(\phi)\subset K_j\}$ is defined by the family $(r_{k,j})_{k\in\mathbb N}$ of seminorms having the form 
$$r_{k,j}(\phi)=\sup\{|\partial^\alpha \phi|\mid |\alpha|\leq k, x\in K_j\}.$$
Fact 3 (see p. 533). A distribution on $(a,b)$ with values in $\mathcal{F}$ is (by definition) a continuous linear map $u$ from $\mathcal{D}(a,b)$ to $\mathcal{F}$.
It follows from facts 1 and 2 that a linear map $u: \mathcal{D}(a,b)\to \mathcal{F}$ is a distribution if:

For every $i\in\mathbb N$ and every $j\in\mathbb N$, there exist $C\geq 0$ and $k_1,...,k_n\in\mathbb N$ such that
  $$p_i(\langle u, \phi\rangle)\leq C\sum_{m=1}^n \sup\{|\partial^\alpha \phi|\mid |\alpha|\leq k_m\}$$
  forall $\phi\in \mathcal{D}(a,b)$ with $\text{supp}(\phi)\subset K_j$.

The above condition is equivalent to the following.

For every $i\in\mathbb N$ and every compact set $K\subset (a,b)$, there exist $C\geq 0$ and $N\in\mathbb N$ such that
  $$p_i(\langle u, \phi\rangle)\leq C\sum_{|\alpha|\leq N} \sup|\partial^\alpha \phi|$$
  forall $\phi\in \mathcal{D}(a,b)$ with $\text{supp}(\phi)\subset K$.

In particular, if $\mathcal{F}=X$ then there is only one seminorm $p_i(\langle u, \phi\rangle)=\|\langle u, \phi\rangle\|$ and thus we get the original definition.
(The OP posted a very similar answer some minutes before me. But as my answer was already written, I decided to post it too.)
