Let $X$ and $Y$ be locally-convex topological vector spaces, with topologies given by families of seminorms $(p_i) _{i \in I}$ and $(q_j) _{j \in J}$, respectively.
If $L : X \to Y$ is a continuous linear operator, is its continuity equivalent to: $\exists p_i \ \exists q_j \ \exists C$ such that $q_j (Lx - Ly) \le C p_i (x - y)$?
I am seeing this in distribution theory quite a lot (for $Y = \Bbb C$) and I would like to know whether this is a general fact in locally-convex spaces (it clearly is in normed spaces).
No, continuity is equivalent to saying that for every $q$ there exist finitely many $p_1,\dots,p_n$ such that $$q(Lx)\le c\sum_{j=1}^np_j(x).$$
You may have been misled by the special case of distributions because in that case there is only one $q$ and any finite collection of $p_j$ is dominated by a single $p$.