$\newcommand\R{\mathbb R}$For real $x_1,x_2,x_3$, let
\begin{equation}
g(x_1,x_2,x_3):=\sin(x_1-x_2),\quad h(x_1,x_2,x_3):=\sin(x_2-x_3).
\end{equation}
The request is to bound
\begin{equation}
P(Y\ge t)
\end{equation}
from above, where
\begin{equation}
Y:=\|w_1 g+w_2 h\|_{L^p}
\end{equation}
and $w_1,w_2$ are independent standard normal random variables.
If $\|\cdot\|_{L^p}$ is understood here as the standard norm on $L^p(\R^3)$, then $Y=\infty$ with probability $1$. This follows because, say, (i) $h(x_1,x_2,x_3)$ is nonzero, continuous, and periodic in $x_3$; (ii) the function $|\cdot|^p$ is continuous and has nonzero variation on any interval of nonzero length; and (iii) $P(w_2\ne0)=1$.
So, for the problem to make sense, assume that the norm $\|\cdot\|_{L^p}$ is the standard norm on $L^p(T)$, where $T$ is a measurable subset of $\R^3$ such that $G:=\|g\|_{L^p(T)}\in(0,\infty)$ and $H:=\|h\|_{L^p(T)}\in(0,\infty)$.
Then, by Minkowski's inequality, for $p\in[1,\infty]$,
\begin{equation}
\begin{aligned}
P(Y\ge t)&\le P(G|w_1|+H|w_2|\ge t) \\
&\le P(G|w_1|\ge\tfrac G{G+H}\,t)+P(H|w_2|\ge\tfrac H{G+H}\,t) \\
&=2P(|w_1|\ge\tfrac t{G+H}) \\
&\le2\exp(-\tfrac{t^2}{2(G+H)^2})
\end{aligned}
\end{equation}
for real $t\ge0$.
It was assumed above that both $G$ and $H$ are $>0$.
If exactly one of the norms $G$ and $H$ is $0$ (and still $G<\infty$ and $H<\infty$), then, similarly, $P(Y\ge t)\le\exp(-\tfrac{t^2}{2(G+H)^2})$ for real $t\ge0$. If $G=H=0$, then, clearly, $P(Y\ge t)=0$ for real $t>0$. So, when the condition that $G>0$ and $H>0$ does not hold, we have better upper bounds than $2\exp(-\tfrac{t^2}{2(G+H)^2})$.
Suppose now that
\begin{equation}
Y:=\|w_1 g_1+\cdots+w_n g_n\|_{L^p},
\end{equation}
where $w_1,\dots,w_n$ are independent standard normal random variables and the $g_j$'s are deterministic functions in $L^p$ with $G:=\max_j\|g_j\|_{L^p}\in(0,\infty)$. Then for $p\in[2,\infty)$, by Theorem 3.3 (with $\Gamma=1$; see also typo correction) and Proposition 2.1, we have the following Bernstein-type bound:
\begin{equation}
P(Y\ge t)\le2\exp\Big(-\frac{t^2}{B^2+t+B\sqrt{B^2+2t}}\Big)
\end{equation}
for real $t\ge0$, where $B:=\sqrt{(p-1)G^2 n}$. In particular, if $t=O(\sqrt n)$ or, more generally, $t=o(n)$ as $n\to\infty$, then
\begin{equation}
P(Y\ge t)\le2\exp\Big(-\frac{t^2}{(2+o(1))B^2}\Big)
=2\exp\Big(-\frac{t^2}{(2+o(1))(p-1)G^2 n}\Big),
\end{equation}
so that we have a normal-type upper bound on $P(Y\ge t)$ with an asymptotically correct constant factor $2+o(1)$ in the denominator of exponent (consider the special case when the $g_j$ is the constant $1$ on (say) the cube $[0,1]^d$, so that one may take here $p=2$).
