I agree with Adam Smith that the question seems a bit misguided, but let me show anyway that the answer is negative away from certain silly cases. Well, first to make a more well-posed question, one first has to adjust the definition of the functor so that it is at least a Zariski-sheaf (ideally without changing the "value" on affines). So let's first do that. Let $E$ be an elliptic curve over a field $k$, and for the directed system of finite extensions $k_i/k$ inside of a fixed separable closure $k_s/k$ let $E_i = {\rm{Res}}_{k_i/k}(E_{k_i})$ denote the corresponding abelian varieties arising by Weil restriction. Define the functor $G = \injlim E_i$ on the category of $k$-schemes. This is an fpqc sheaf on the category of affine $k$-schemes, since we can explicitly compute it: for any $k$-algebra $R$, $G(R) = \injlim E(R_{k_i}) = E(R_{k_s})$, the final equality using that the functor of $E$ commutes with the formation of direct limits in $k$-algebras (thanks to Grothendieck's awe-inspiring necessary and sufficient functorial characterization of locally finitely presented morphisms of schemes, from EGA IV$_3$, 8.14, which we will use in a more impressive converse direction shortly).
By inspection, the functor $G$ satisfies the fpqc sheaf axioms on affines. Thus, if we Zariski-sheafify $G$ on the category of $k$-schemes then the resulting functor $G'$ has the same value on affines and so is easily seen to be an fpqc sheaf. So the "right" question is whether $G'$ is representable. I will now prove the "expected" negative answer whenever $k$ is not separably closed or real-closed, which is to say (by the Artin-Schreier theorem) that $k_s/k$ has infinite degree. (Obviously need to rule out the cases when $[k_s:k]$ is finite!)
Suppose $G'$ is represented by a $k$-scheme (which we will also denote by $G'$), so by Grothendieck's functorial criterion we see that $G'$ is locally of finite type over $k$. Thus, the identity component ${G'}^0$ makes sense and it finite type (as for any locally finite type group scheme over a field; see SGA3, VI$_{\rm{A}}$, 2.4). As such, it contains the connected $E_i$ as closed $k$-subgroup schemes since any monomorphic homomorphism between finite type group schemes over a field is a closed immersion (proved in SGA3, VI$_{\rm{B}}$, 1.4.2, for example). But $E_i$ has dimension $[k_i:k]$, which grows without bound since $[k_s:k]$ is infinite by hypothesis. Hence, ${G'}^0$ contains closed subschemes of arbitrarily large dimension, an absurdity since ${G'}^0$ is finite type over a field. QED