It doesn't seems true to me. Take an orthonormal sequence of continuous functions $\{\phi_n\}_{n\in\mathbb{N}}$ on $[0,1]$ with diverging $L^\infty$ norm. For instance $$\phi_n:=2^{n/2}f(2^n x)$$
with a continuous function $f$ , with $\|f\|_ 2=1 $, and $\operatorname{supp}(f)\subset [1/2,1]$ (so the $\phi_n$ have disjoint supports). They are eigenfunctions of a continuous, symmetric, non-negative kernel, $$K(x,y):=\sum_{n=0}^\infty 3^{-n}\phi_n(x)\phi_n(y)\\ . $$.