Tikz draw contour without some edges, and fill

With use TikZ library pgfplots.fillbetween

\documentclass[margin=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usetikzlibrary{pgfplots.fillbetween}

\begin{document}
    \begin{tikzpicture}
\draw[name path=A]  (0, 0.3) to [out=0,in=180] ++(5,-0.6);
\draw[name path=B]  (0,-0.3) to [out=0,in=180] ++(5, 0.6);
\tikzfillbetween[of=A and B] {fill=gray!30};
    \end{tikzpicture}
\end{document}

Quick and dirty: fill the area first and then draw what you want to draw. It is not easily possible to switch off the drawing of a part of a path (but it is possible yet considerably more effort).

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
  \useasboundingbox (-1,-1) rectangle (6,2);
  \path[fill=black!10] (0,1) to[out=0,in=180] ++(5,-0.6) -- ++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
  \draw (0,1) to[out=0,in=180] ++(5,-0.6)++(0,0.6) to[out=180,in=0] ++(-5,-0.6);
\end{tikzpicture}
\end{document}

You can switch the path on and off e.g. using this answer.

\documentclass[tikz]{standalone}
\pgfkeys{tikz/.cd,
edge options/.code={\tikzset{edge style/.style={#1}}},
}
\begin{document}
\begin{tikzpicture}[every edge/.append code = {% https://tex.stackexchange.com/a/396092/121799 
    \global\let\currenttarget\tikztotarget % save \tikztotarget in a global variable
    \pgfkeysalso{append after command={to[edge style] (\currenttarget)}}},
every edge/.append style={edge style}   ] 
  \useasboundingbox (-1,-1) rectangle (6,2);
  \path[fill=black!10] 
  (0,1) [edge options={out=0,in=180,draw=black}]  edge  ++(5,-0.6) 
  -- ++(0,0.6) 
  [edge options={out=180,in=0,draw=black}] edge ++(-5,-0.6);
\end{tikzpicture}
\end{document}

For comparison, with plain Metapost, where filling and drawing are kept separate, and where you can concatenate path variables directly.

\documentclass[border=5mm]{standalone}
\usepackage{luatex85}
\usepackage{luamplib}
\begin{document}
\begin{mplibcode}
beginfig(1);
    z0 = (-72, 8.5);

    path A, B;
    A = z0 {right} .. {right} z0 rotated 180;
    B = A reflectedabout(up, down);

    fill A -- B -- cycle withcolor 7/8[blue, white];
    draw A;
    draw B;
endfig;
\end{mplibcode}
\end{document}

(compile with lualatex...)