Does taking Vicodin at the same time as Zoloft lessen the
effectiveness of taking Zoloft for depression?

If anything, taking the two at the same time should enhance the
effects of both, since vicodin is 99% bound to plasma protein and
zoloft is 98% bound, and they will cause each other to dissociate from
the protein, thus being more available to the site of action. Also, I
can't see the two interacting (except with liver enzymes, see below)
since one is an opioid and the other a serotonin reuptake inhibitor.
In fact, serotonin neurons in the spinal cord can cause release of
enkephalins, which decrease pain, so again, I can only see it
enhancing the effect. Both drugs use the liver enzymes CYP2D6, so that
also will cause more of each drug to be available to the body.
Overall, the two drugs should significantly enhance each other.

I would consult my pharmacist.  It's a free consultation and a safe
and free way to answer your question.