Please find below my suggestion about your problem.

buildLeft[randList_, ratorList_] :=  
  Append[Flatten@
    Transpose[{Take[randList, Length@randList - 1], ratorList}] , 
   Last@randList];

leftEqBuilder[randList_, ratorList_] :=  
  StringJoin[
   If[MemberQ[randList, #], ToString@#, #] & /@ 
    buildLeft[randList, ratorList]];

solver[randList_, ratorList_, rhs_] := 
  Module[{ tup = Tuples[ratorList, Length[randList] - 1]},
   leftEqBuilder[randList, 
    SelectFirst[tup, 
     ToExpression[leftEqBuilder[randList, #]] == rhs &]]];

solutionQ[str_, rhs_] := ToExpression@str == rhs;

Tests

solver[{1, 2, 3, 4}, {"+", "-"}, 10]
(*      1+2+3+4                *)


solver[{75, 4, 10, 6, 2, 2}, {"+", "-", "*", "/"}, 286]
(*   75*4+10-6*2*2                                          *)

Check

solutionQ[solver[{1, 2, 3, 4}, {"+", "-"}, 10],10]
(*  True *)
solutionQ[solver[{75, 4, 10, 6, 2, 2}, {"+", "-", "*", "/"}, 286], 286]
(* True *)

Note that the solution provided is quite general: The operands do not have to be numbers. The operators need only to be compatible with the operands. Thanks,

The answer to you question is YES. You need just to add a on-liner parser to accomplish what you want. I have added such a parser and the new solver. Please find below the updated code.

buildLeft[randList_, ratorList_] :=  
  Append[Flatten@
    Transpose[{Take[randList, Length@randList - 1], ratorList}] , 
   Last@randList];

leftEqBuilder[randList_, ratorList_] :=  
  StringJoin[
   If[MemberQ[randList, #], ToString@#, #] & /@ 
    buildLeft[randList, ratorList]];

solver[randList_, ratorList_, rhs_] := 
  Module[{ tup = Tuples[ratorList, Length[randList] - 1]},
   leftEqBuilder[randList, 
    SelectFirst[tup, 
     ToExpression[leftEqBuilder[randList, #]] == rhs &]]];
solutionQ[str_, rhs_] := ToExpression@str == rhs;
parseExpression[expr_, dum_: "?"] := 
  Module[{s = 
     StringSplit[StringReplace[expr, " " -> ""], "="]}, {StringSplit[
     First@s, dum], Last@s}];
newSolver[strExpr_, ratorList_, dum_: "?"] := 
  Module[{s = parseExpression[strExpr, dum]}, 
   solver[ToExpression@HoldForm /@ First@s, ratorList, 
     ToExpression@Last@s] <> " = " <> Last@s];    

Now let's test it.

newSolver["75*4?10?6?2*2=286", {"+", "-", "*", "/"}]
(*  75*4+10-6*2*2 = 286   *)   
newSolver["1?1?1=3", {"+", "-", "*", "/"}]
(* 1+1+1 = 3 *)

Thanks,