You can do this with the following script attached to the spreadsheet that receives form submissions.
Distributing among sheets within the same spreadsheet
For simplicity, I assume that the submissions go to the first sheet of the spreadsheet (the master), and they should be randomly placed into one of the other sheets. This is what the script below does; to use it, enter it in Tools > Script Editor and add a trigger (Resources > This project's trigger) with parameters "From spreadsheet, on form submit".
function distribute(e) {
var ss = SpreadsheetApp.getActiveSpreadsheet();
var sheets = ss.getSheets();
var sheet = sheets[1+Math.floor(Math.random()*(sheets.length-1))];
var height = e.range.getHeight()
var width = e.range.getWidth()
var lastRow = sheet.getLastRow();
sheet.insertRowsAfter(lastRow, height);
e.range.copyTo(sheet.getRange(lastRow+1, 1, height, width));
}
An alternative approach to sheet selection is to have a list of specific sheets to which the content should be placed. If this is preferred, replace the lines with var sheets = ... var sheet = ...
by the following:
var sheets = ['Sheet A', 'Sheet B', 'Sheet C'];
var sheet = ss.getSheetByName(sheets[Math.floor(Math.random()*sheets.length)]);
Distributing among different spreadsheets
Here I'm assuming the data goes to the first sheet of each spreadsheet.
function distribute(e) {
var ss = SpreadsheetApp.getActiveSpreadsheet();
var arrayss = ['Id1', 'Id2', 'Id3']; // Ids of other spreadsheets
var targetss = SpreadsheetApp.openById(arrayss[Math.floor(Math.random()*arrayss.length)]);
var sheet = targetss.getSheets()[0];
var height = e.range.getHeight()
var width = e.range.getWidth()
var lastRow = sheet.getLastRow();
sheet.insertRowsAfter(lastRow, height);
sheet.getRange(lastRow+1, 1, height, width).setValues(e.range.getValues());
}