A space of the form $C_p(X)$ (the space of continuous functions on a Tychonoff space $X$, in the pointwise-topology (i.e. the subspace topology induced from the product topology on $\mathbb{R}^X$) is often an example; note that such spaces are locally convex spaces:

$C_p(X)$ is separable when $X$ is second countable, but $C_p(X)$ is not second countable unless $X$ is countable. $C_p(X)$ is Lindelöf when $X$ is second countable and metrisable. (Proofs can be found in Arhangel'skij's book on $C_p(X)$ or Tkachuk's series of books on them etc.) So concretely, $C_p(\mathbb{R})$ or $C_p([0,1])$ is an example of a Lindelöf, separable locally convex space that is not second countable. (It does have countable network weight, as $nw(X) = nw(C_p(X))$ for all $X$).