Please try:

=vLOOKUP(R3,Calculations!$B$2:$C$7,2,0)

For LOOKUP (as opposed to VLOOKUP) to work properly the data must be sorted.

The anchors ($) are in case say the formula was to be copied down (eg to return a value from R4). Without them the array range would slide down automatically until by R9 the search would be of Calculations!B8:C13.

Yes, it is possible to emulate many of the built-in functions by using Class Spreadsheet (SpreadsheetApp) and JavaScript Array Object and its methods, but "the emulations" usually will be slower than built-in functions.

Knowing the above I consider that the core of the question is how to use JavaScript to emulate VLOOKUP and other built-in functions.

I order to keep this answer short, here is an example taken from the answer to Writing google Javascript similar to vlookup

function findinB() {
  var sh = SpreadsheetApp.getActiveSheet();
  var ss = SpreadsheetApp.getActiveSpreadsheet();
  var last=ss.getLastRow();
  var data=sh.getRange(1,1,last,2).getValues();// create an array of data from columns A and B
  var valB=Browser.inputBox('Enter value to search in B')
  for(nn=0;nn<data.length;++nn){
    if (data[nn][1]==valB){break} ;// if a match in column B is found, break the loop
      }
Browser.msgBox(data[nn][0]);// show column A
}

Just replace the Browser.inputBox by the data sources that you want to use.

Note: There are several questions on Stack Overflow about how to implement VLOOKUP on JavaScript and/or Apps Script, like Does JavaScript or jQuery have a function similar to Excel's VLOOKUP?

Try using SUMIF (Documentation) instead of LOOKUP or VLOOKUP.

Like so:

=SUMIF(D5:D16;"tax";E5:E16)

Also, as you are using a Polish "Locale", you should use ; instead of ,.

Reasoning: whether you have to use , or ; depends on the "Locale" of the spreadsheet, under File –> Spreadsheet Settings. For example, a US or UK "Locale" would use the former (,) while a Hungarian or Polish "Locale" would use the latter (;)