Would the proof still work if used $[\lor_{R_1}]$ and $[\lor_{R_2}]$ before $[∃_R]$, in both branches.
I assumed appluing $[∃_R]$ to $B$ and $A$ in the respective branches would be redundant since both would be eliminated in the respective branches by $[\lor_{R_1}]$ and $[\lor_{R_2}]$
No, you can not use thee rules $\lor_{R_1}$ or $\lor_{R_2}$ before the rule $\exists_R$, in either branch. (I assume you use the word "before" reading a derivation _bottom-up-, thus "before" amounts to "below").
Indeed, in the OP's derivation, the conclusion of the rule $\exists_R$ is the sequent $B[x\backslash y] \vdash \exists x (A \lor B)$. On the right of the turnstile symbol $\vdash$, the formula is $\exists x (A \lor B)$, and then you cannot apply any rule $\lor_{R_i}$, because $\lor$ is not the main connective of that formula.
You could apply a rule $\lor_{R_i}$ if the formula on the right-hand side of the sequent were $(\exists x A) \lor B$, but this is not the case.