Rustington, LIttlehampton, Chichester
The Sidlesham entry Harters mentioned is the Crab and Lobster, for which I can strongly vouch, as it's a few doors along from the in laws. Beautiful old pub overlooking the marsh, now had the full Mr and Mrs Smith treatment. Great on seafood, but I also love their (so tender you can eat it with a teaspoon) ox cheek.
No idea whether it will be open on NYD, but worth a call, and a lovely place to walk the lunch off afterwards. It's about 10 mins south of Chichester, follow signs for Selsey and Pagham Harbour.
The East Beach Café has unusually good food for a small beach place. We enjoyed a nice lunch there several years ago, but it was during the summer. How that area seems in Winter, I don't know, but often it's quite warm then. The fish and chips looked more than OK. As Harters suggested, many places are closed that day so check for anywhere you decide to eat.
Chichester is an interesting town but I'd think most places will be closed for lunch, but who knows.
I'm not really familiar with the area west of Worthing - but both the Good Food Guide and Hardens list places at Tangmere, Sidlesham, East Wittering as well as a handful in Chichester. Hardens also has the East Beach Cafe in Littlehampton - its website suggests it may tick the seafood box but no idea if they are open on NYD.