Hi Pinocchio.......

One of the biggest producers on the Island is the Callinico Winery..

http://callinico.myrties.com.gr/index_en.php

Nothing on the pages of the website about tours but, i'm sure you can tour the cellars and taste the wines there.

Alternatively there are various day trips that include visits to some of the vineyards on the island.
Zakfari used to make a stop at one but, i couldn't see it listed on the itinerary of the Tsilivi travel page.
http://www.tsilivi-travel.gr/zakynthos.zante.zakfari.jeep.php

Might be worth giving them al call.. .. Or ask at your Hotel / apartments. A few years back our family hotel took us out to their own humble vineyard for a taster and look around, a trip we would have missed out on had we not asked the very same question you put here.. ..

Thanks for the info nut.magnet

When in Zante 1997 ( Ages ago) we did try the Callinico wine and it was vgood . Cant remember the exact type but it was vey heart warming. Havent had it since as weve not been back to zante.

On the Callinico link i posted it does actually mention tours..

Just above where they start to list the grape types...

http://callinico.myrties.com.gr/index_en.php

I think they have lost their web site account.. maybe all the tour info was on there..

http://www.callinicowines.gr/