  In this work, we undertake an analysis of new wormhole solutions within an
action-dependent Lagrangian framework. These geometries can be traversable and
supported by a positive energy density. The modification of the gravitational
field equations is produced by the inclusion in the gravitational Lagrangian
linear of a background four-vector $\lambda_{\mu}$. This new term expands
significantly the conventional description of gravity making it highly
non-linear, and therefore drawing general conclusions about legitimate forms of
$\lambda_{\mu}$ proves a formidable task in general. It is, then, customary to
adopt an ansatz that strikes a balance between enabling new phenomenology while
retaining a significant degree of generality on $\lambda_\mu$. Ours is given by
the choice $\lambda_\mu=(0,\lambda_1(r), 0, 0)$, with an arbitrary
$\lambda_1(r)$. By setting $\lambda_1(r)=-1/r$ we craft new families with
physically desirable properties, but the wormholes thus generated turn out to
be conical, as evidenced by an angle deficit, in a similar fashion to other
known solution families. Under the general shape of $\lambda_1(r)$, we
demonstrate that these solutions are not compatible with the Null Energy
Condition (NEC) in general, as it happens to their General Relativity
counterparts, except on specific occasions where the derivative of the redshift
function of the metric diverges at the throat (however, in these latter cases,
the traversability of the wormhole will be disrupted). On the other hand, it is
possible to solve the conical character and satisfies the flatness condition
for more general functions of $\lambda_1(r)$.
