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I think the answer is in general no, because of the following example.

Take $10$ lines in $\mathbb{P}^2$ all passing through a point $p$, and let $D$ be their union. There exist a covering $Y \to \mathbb{P}^2$ of degree $5$ branched on $D$, and $p$ is the only point of total ramification. The only singularity of $Y$ is the unique preimage $v$ of $p$.

In order to understand the singularity of $Y$ at $v$, let us consider the blow-up $\widehat{\mathbb{P}}$ of $\mathbb{P}^2$ at the point $p$; we obtain an exceptional curve $E$, and by the Riemann-Hurwitz formula we see that the preimage of $E$ in the induced cover $\widehat{Y} \to \widehat{\mathbb{P}}$ is an elliptic curve with self-intersection $(-5)$.

From this we recognize that $Y$ is nothing but the elliptic normal cone in $\mathbb{P}^5$ (that is, the cone over the elliptic normal curve of degree $5$ in $\mathbb{P}^4$) and $f \colon Y \to \mathbb{P}^2$ is a general projection to $\mathbb{P}^2$.

Now, Saito (Einfach-elliptische Singularitaten, 1974) proved that the singularity at the vertex of an elliptic normal cone (a so-called simple elliptic singularity) is a complete intersection if and only if the degree of the cone is $\leq 4$.

In this case the degree is $5$, so $Y$ is not a local complete intersection.

All the stuff about the covering obviously works in any characteristic (well, maybe different from $5$), but I think Saito's result requires the ground field to be $\mathbb{C}$, so I do not know whether this works also in the case of positive characteristic.

EDIT As pointed out by Angelo, in this example $D$ is not simple normal crossing, so it does not really answer the question. Anyway, I have let me try to give another example which works with normal crossing $D$.

Take the product $X=E \times F$ of two elliptic curves, and consider the divisor $D=E_1+E_2+F_1+F_2$, where the notation is obvious. Now $D$ has only nodal singularities, so it is simple normal crossing. Moreover, by using Pardini's construction explained in "Abelian covers of algebraic variety"varieties", it is possible to construct a $\mathbb{Z}_3$-cover $Y \to X$ whose singularities are

• two rational double points of type $A_2$ and
• two points of type $1/3(1,1)$

(these singularities obviously lie over the four singular points of $D$).

The points of type $1/3(1,1)$ are locally analitycally a cone over a twisted cubic curve in $\mathbb{P}^3$, so they are not complete intersection singularities.

I think the answer is in general no, because of the following example.

Take $10$ lines in $\mathbb{P}^2$ all passing through a point $p$, and let $D$ be their union. There exist a covering $Y \to \mathbb{P}^2$ of degree $5$ branched on $D$, and $p$ is the only point of total ramification. The only singularity of $Y$ is the unique preimage $v$ of $p$.

In order to understand the singularity of $Y$ at $v$, let us consider the blow-up $\widehat{\mathbb{P}}$ of $\mathbb{P}^2$ at the point $p$; we obtain an exceptional curve $E$, and by the Riemann-Hurwitz formula we see that the preimage of $E$ in the induced cover $\widehat{Y} \to \widehat{\mathbb{P}}$ is an elliptic curve with self-intersection $(-5)$.

From this we recognize that $Y$ is nothing but the elliptic normal cone in $\mathbb{P}^5$ (that is, the cone over the elliptic normal curve of degree $5$ in $\mathbb{P}^4$) and $f \colon Y \to \mathbb{P}^2$ is a general projection to $\mathbb{P}^2$.

Now, Saito (Einfach-elliptische Singularitaten, 1974) proved that the singularity at the vertex of an elliptic normal cone (a so-called simple elliptic singularity) is a complete intersection if and only if the degree of the cone is $\leq 4$.

In this case the degree is $5$, so $Y$ is not a local complete intersection.

All the stuff about the covering obviously works in any characteristic (well, maybe different from $5$), but I think Saito's result requires the ground field to be $\mathbb{C}$, so I do not know whether this works also in the case of positive characteristic.

EDIT As pointed out by Angelo, in this example $D$ is not simple normal crossing, so it does not really answer the question. Anyway, I have another example which works with normal crossing $D$.

Take the product $X=E \times F$ of two elliptic curves, and consider the divisor $D=E_1+E_2+F_1+F_2$, where the notation is obvious. Now $D$ has only nodal singularities, so it is simple normal crossing. Moreover, by using Pardini's construction explained in "Abelian covers of algebraic variety", it is possible to construct a $\mathbb{Z}_3$-cover $Y \to X$ whose singularities are

• two points of type $A_2$ and
• two points of type $1/3(1,1)$

(these singularities obviously lie over the four singular points of $D$).

The points of type $1/3(1,1)$ are locally analitycally a cone over a twisted cubic curve in $\mathbb{P}^3$, so they are not complete intersection singularities.

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