Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme
There is no reason in Shamir's scheme for the finite field $\mathbb F$
to have a prime number $p$ of elements; the field can have $p^m$ elements for 
suitable prime $p$ and integer $m \geq 1$.
So, using $F_{2^8}$, the field with $2^8$ elements is perfectly all right.
However, choosing $m = 1$ has the advantage that calculations in $\mathbb F_p$
can be done using the
standard arithmetic unit included in microprocessors and the like (followed
by integer divisions for doing the mod $p$ operations) whereas using 
$\mathbb F_{p^m}$ requires having a library already available (or developing
one) or building a processor for arithmetic in the field.

Given that you have the capability of doing arithmetic in $\mathbb F_{2^8}$,
you can use this field if you like. But, (as you say) in Shamir's
scheme, the secret is just one element of the field (one $8$-bit byte), 
and so, if the Secret to be shared is several bytes long, you will
need to process each byte separately into its (one-byte) shares, and 
concatenate the $i$-th share bytes into the $i$-th Share of the Secret.
Keep in mind that each byte  can be divided into at most $255$ shares
(the point $q(0)$ on $q(x)$ cannot be used for obvious reasons),
and so if you need to have more than $255$ shares to distribute, you
will have to use a different field. Finally, to maintain cryptographic
security, as the shares corresponding to each byte of the Secret are
being computed, you should do exactly what you say you are doing
in your comment: 
use a different set of "randomly chosen coefficients" for $q(x)$, 
rather than re-use the same set over
and over again for finding the shares of all the bytes.