Biggest in the area is Dickie's of Bangor. Usually plenty on brokerage and, weather permitting, a beautiful area to visit.
Smaller chandlers and yards in Porthmadog, Pwllheli, Abersoch, Caernarfon, Holyhead, Conwy, Deganwy, Rhyl.
Personally I would miss them all except Dickie's and look at some of the spectacular places. South Stack at mid ebb, Menai Bridge/Swellies, Bardsey Sound from the mainland, especially in a blow with the tide against it, Snowdonia. Look <A target="_blank" HREF=http://www.fachwen.org/snowdonia/index.html>here</A> for some more ideas.
As Vyv says, Dickies at Bangor is big, but there isn't much else in Bangor to look at. You will enjoy Pwllheli for the marina, views and two good chandlers/yards - Firmhelm has a modern largish outlet and Partington Marine has a very comprehensive stock with good clothing etc in a traditional boatyard setting.
In Abersoch there is Abersoch Boatyard which has a large clothing stock as well as routine chandlery.
If you want a special view of some dangerous water and also take in Bardsey Isalnd, go to the tip of the Llyn peninsula to the old coastguard lookout station near Aberdaron.
There are also boat trips to Bardsey from there, an outing which is well recommended.