[RPG] What Paladin fighting styles are usable while wild shaped

All of the Fighting Styles available to the Paladin either require you to be wielding a weapon in one hand, wielding a weapon in both hands, wielding a shield, or wearing armour, so by default, you can't use any of them while Wild Shaped. A generous DM might let you use a weapon or shield while Wild Shaped into something like an Ape, but I wouldn't.

However, you should be able to take advantage of the barding rules:

Barding. Barding is armor designed to protect an animal’s head, neck, chest, and body. Any type of armor shown on the Armor table in this chapter can be purchased as barding. The cost is four times the equivalent armor made for humanoids, and it weighs twice as much.

Then, when you Wild Shape into the animal, one of your allies can put the barding on you. This should allow you to gain the benefits of the Defense Fighting Style.

There is another possibility - if your DM allows you to ignore the (entirely arbitrary) restrictions on which Fighting Styles the Paladin can choose, you could take the Archery Fighting Style. It says:

You gain a +2 bonus to attack rolls you make with ranged weapons.

So if you Wild Shape into an animal that has a ranged weapon attack (like the Ape, for example), the Archery Fighting Style should still function. It's definitely worth asking your DM about.