$\newcommand\R{\mathbb R}\newcommand{\al}{\alpha}\newcommand{\de}{\delta}\newcommand{\J}{\mathcal J} \newcommand{\be}{\beta}\newcommand{\ep}{\varepsilon}$Yes, such a construction exists.
Indeed, take any real $\ep>0$. For a real $C\ge0$ and $\al\in(0,1]$, let us say that a function $g$ is $(C,\al)$-Hölder on a set $S\subseteq[0,1]$ if $|g(y)-g(x)|\le C|y-x|^\al$ for all $x,y$ in $S$.
Without loss of generality (wlog), the function $f$ is $(1,\al)$-Hölder on $[0,1]$.
Let $Z:=\{z\in I:=[0,1]\colon f(z)=0\}$ and $N:=I\setminus Z$. Wlog, $\{0,1\}\in Z$ (otherwise, extend $f$ appropriately to an interval $[A,B]\supset I$ so that $f(A)=0=f(B)$ and then shrink the interval $[A,B]$ to $I$). So, $N=\bigcup_{J\in\J}J$ for some (countable) set $\J$ of pairwise disjoint nonempty open subintervals of $I$.
It is enough to construct a smooth function $f_\ep$ such that for each $J\in\J$ \begin{equation*} \|f-f_\ep\|_{C^\be(J)}\le L(\al,\be)\ep \tag{10}\label{10} \end{equation*} and \begin{equation*} |f_\ep|\le|f|\text{ on } J, \tag{20}\label{20} \end{equation*} where $L(\al,\be)$ is a real number depending only on $\al,\be$.
To begin such a construction, take any real $\de>0$. Let $\J_\de$ denote the set of all intervals $J\in\J$ of length $>4\de$. Of course, the set $\J_\de$ is finite. Let \begin{equation*} f_\ep(x):=0\text{ for }x\in I\setminus\bigcup_{J\in\J_\de}J. \end{equation*}
If for each interval $(a,b)\in\J_\de$ we have \begin{equation} \text{$f_\ep:=0$ on $(a,a+\de)\cup(b-\de,b)$ }\tag{30}\label{30} \end{equation} and $f_\ep$ is smooth on $(a,b)$, then $f_\ep$ will be smooth on the entire interval $I$.
Take now any interval $(a,b)\in\J_\de$ indeed. It is enough to construct a function $f_\ep$ which is smooth on $J$ and such that \eqref{10}, \eqref{20}, and \eqref{30} hold.
Note that (i) $f(a)=0=f(b)$ and (ii) either $f>0$ on $(a,b)$ or $f<0$ on $(a,b)$. Wlog, $f>0$ (on $(a,b)$). Moreover, $f$ is continuous. So, \begin{equation*} f\ge2c\text{ on }[a+\de,b-\de] \tag{40}\label{40} \end{equation*} for some real $c>0$. For a very small $\eta\in(0,\de]$, let $K_\eta$ be a smooth nonnegative function supported on the interval $[-\eta,\eta]$ such that $\int K_\eta=1$. Let then \begin{equation*} g_\eta:=g*K_\eta, \end{equation*} where \begin{equation*} g:=(f-c)1_{[a+2\de,b-2\de]}. \end{equation*}
Then, by \eqref{40}, $g\ge0$ and hence, in view of conditions $\eta\in(0,\de]$ and \eqref{40},
\begin{align}
g_\eta(x)&=\int_{-\de}^\de du\,K_\eta(u)(f(x-u)-c)1(x-u\in[a+2\de,b-2\de]) \tag{45}\label{45} \\
&\le\eta^\al+\int_{-\de}^\de du\,K_\eta(u)(f(x)-c)1(x-u\in[a+2\de,b-2\de]) \notag \\
&\le\eta^\al+(f(x)-c)1(x\in[a+\de,b-\de]) \notag \\
&\le\eta^\al+(f(x)-c)\le f(x) \notag
\end{align}
for $x\in(a,b)$
if
\begin{equation*}
\eta\in(0,\min(\de,c^{1/\al})), \tag{50}\label{50}
\end{equation*}
and \eqref{50} will be henceforth assumed.
Also, the integrand in \eqref{45} is $\ge0$ for all $x\in(a,b)$ and $=0$ for $x\in(a,a+\de)\cup(b-\de,b)$. So, $g_\eta\ge0$ on $(a,b)$ and $g_\eta=0$ on $(a,a+\de)\cup(b-\de,b)$.
So, letting
\begin{equation*}
f_\ep:=g_\eta\text{ on }J=(a,b),
\end{equation*}
we see that $f_\ep$ is smooth on $J=(a,b)$, and conditions \eqref{20} and \eqref{30} hold.
So, it remains to check that condition \eqref{10} for our interval $J=(a,b)$ holds if $\de$ and $c$ are chosen to be small enough. This is rather straightforward and will be detailed later. (It is rather late here now.)