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,

Would it be possible to use an infix like "?" (or something else) as new custom operator and call the above solver function?

The Wolfram Alpha built-in Infix function is not designed to define custom operator right?

(like custom-defined infix operators in Perl 6)

Bottom line it would be nice for Wolfram Alpha to guess that ? is for the missing numeric operators.

in: 1 ? 1 ? 1 = 3
out: 1 + 1 + 1 = 3

https://www.wolframalpha.com/input/?i=1+%3F+1+%3D+2

Extend the InputForm capabilities to understand the meaning of ? or a create a special sign for this purpose

REFERENCES

• https://reference.wolfram.com/language/ref/Infix.html

• https://reference.wolfram.com/language/ref/InputForm.html?q=InputForm

• https://reference.wolfram.com/language/guide/WolframLanguageSyntaxCharacters.html

• https://reference.wolfram.com/language/guide/MathematicalNotationCharacters.html

• https://docs.perl6.org/language/operators

• http://code.activestate.com/recipes/384122/