I'm planning for a trip to Puerto Rico for about 6 days. I will probably stay in Fajardo most of the time. What is the best accommodation that has a kitchen so that I can cook my meals? Also are the seafood markets in Fajardo good? I've been googling about this but couldn't find anything.
I would suggest renting a room with Airbnb. Very often hosts allow guests to use the kitchen. This is usually specified in the listing description. A quick search for Fajardo reveals 500+ rental offers (rooms and apartments) meaning you have plenty of choice location-wise.
As a corollary to this idea you could rent a full flat thus having the guarantee of a kitchen at your disposal. This of course might be a more expensive solution.
According to hostels.com and hostelworld.com, there is at least one hostel in Fajardo, Puerto Rico, called the Moon Bay hostel. Their list of facilities includes "Full kitchen with brand new appliances!" so with little doubt you can buy your own food and cook it at the hostel. I haven't seen any picture of the kitchen among the pictures shown though, just a mention of it in reviews.
The reviews about this place are interesting. The hostel is actually located near the ferry terminal, but not close to town - and therefore not close to food markets. I'd recommend you to read these reviews before going there.
I did not really understand your second question, however: are you asking if the food sold in seafood markets is edible, or good like recommended gourmet good? In both cases, I would say that if there are running businesses, people buy - and eat - the food. So it is "good".