emtom has found the right reference, but there is a more explicit result in that book (Ireland and Rosen, A Classical Introduction to Modern Number Theory). In fact, Theorem 5 of Chapter 8 (on page 103) directly implies that the number $N = N_{p,\alpha}$ of solutions to $x^4+y^4=\alpha$ in $\mathbb{F}_p$ satisfies the inequality $$ \left| N - p \right| \leq M_0 + M_1 p^{1/2} $$ for some $M_0$ and $M_1$ that are described explicitly in the statement of the theorem (and from that description it is quite easy to see that they do not depend on $p$ or $\alpha$).

Of course, this is no better (and possibly slightly worse) than what you get from the Hasse-Weil bound which Dan Loughran referred to, but at least this reference has the virtue of providing a completely elementary proof.

emtom has found the right reference, but there is a more explicit result in that book (Ireland and Rosen, A Classical Introduction to Modern Number Theory). In fact, Theorem 5 of Chapter 8 (on page 103) directly implies that the number $N = N_{p,\alpha}$ of solutions to $x^4+y^4=\alpha$ in $\mathbb{F}_p$ satisfies the inequality $$ \left| N - p \right| \leq M_0 + M_1 p^{1/2} $$ for some $M_0$ and $M_1$ that are described explicitly in the statement of the theorem (and from that description it easily follows they can be bounded in a way that is independent of $p$ or $\alpha$).

It then automatically follows that for sufficiently large $p$, we will have $N_{p,\alpha}>0$ for all $\alpha$. In other words, for sufficiently large $p$, the expression $x^4+y^4$ assumes all values of $\mathbb{F}_p$ as $x$ and $y$ run through $\mathbb{F}_p$.

Of course, this is no better (and possibly slightly worse) than what you get from the Hasse-Weil bound which Dan Loughran referred to, but at least this reference has the virtue of providing a completely elementary proof.