Is it necessary to choose a primitive polynomial for an S-Box?
Actually, it is not necessary (and, as the polynomial they actually use in AES, $x^8 + x^4+ x^3 + x + 1$, is not primitive, and so it's a good thing that it's not necessary).
The polynomial must be irreducible (if it isn't, the multiplication operation isn't invertible in general, and hence you don't get a field); however it can be shown that the selection of the polynomial doesn't actually affect the security of the cipher; for any irreducible polynomial, you can adjust the affine transformation within the sbox, the mix column transformation, and the rcon constants within the key schedule to come up with an equivalent cipher (equivalent in the sense that an attack on the modified cipher translates into an attack on the original AES).
In other words, Is it necessary for the feild to be of full order(255) for a S-Box?
If you mean that there must be a field element with (multiplicative) order 255; well, yes, any representation of $GF(2^8)$ will have precisely 128 such elements. However, a polynomial being primitive will imply that the element with representation
02 will have such an order; with the standard AES representation, it doesn't.