Comment promoted to answer:

The following is Corollaire 1.5 in SGA5, Exposé VII:

Let $Z$ be any scheme, $p:X\to Z$ a rank $r$ vector bundle, $U=X-Z$, and $q:U\to Z$ the retsriction of $p$. Let $n$ be coprime to the residual characteristics of $Z$ and let $L$ be a sheaf of $\mathbb{Z}/n$-modules on $Z$. Then there is a long exact sequence

 

$$\to H^{\ast-2r}(Z,L(-r))\to H^\ast(Z,L)\to H^\ast(U,q^\ast L)\to H^{\ast+1-2r}(Z,L(-r))\to$$

That's how the corollary is stated but they also show that the sequence is naturally isomorphic to the more familar-looking sequence for cohomology with supports

$$\to H^{\ast}_Z(X,p^\ast L)\to H^\ast(X,p^\ast L)\to H^\ast(U,q^\ast L)\to H^{\ast+1}_Z(X,p^\ast L)\to$$

You should definitely have a look at the rest of exposé VII in SGA5, which proves a lot of related "expected" theorems for étale cohomology with very mild hypotheses (compared to what can be found elsewhere).

Comment promoted to answer:

The following is Corollaire 1.5 in SGA5, Exposé VII:

Let $Z$ be any scheme, $p:X\to Z$ a rank $r$ vector bundle, $U=X-Z$, and $q:U\to Z$ the retsriction of $p$. Let $n$ be coprime to the residual characteristics of $Z$ and let $L$ be a sheaf of $\mathbb{Z}/n$-modules on $Z$. Then there is a long exact sequence

$$\to H^{\ast-2r}(Z,L(-r))\to H^\ast(Z,L)\to H^\ast(U,q^\ast L)\to H^{\ast+1-2r}(Z,L(-r))\to$$

That's how the corollary is stated but they also show that the sequence is naturally isomorphic to the more familar-looking sequence for cohomology with supports

$$\to H^{\ast}_Z(X,p^\ast L)\to H^\ast(X,p^\ast L)\to H^\ast(U,q^\ast L)\to H^{\ast+1}_Z(X,p^\ast L)\to$$

You should definitely have a look at the rest of exposé VII in SGA5, which proves a lot of related "expected" theorems for étale cohomology with very mild hypotheses (compared to what can be found elsewhere).