This plugin is an old plugin and it does not work with the latest version of DataTables.

But we can take a JavaScript function that calculates similarity and use it in a custom search through the SearchBuilder extension.

First, copy this JavaScript code and save it under the name levenshtein.js:

/*
BSD 2-Clause License

Copyright (c) 2018, Tadeusz Łazurski
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

* Redistributions of source code must retain the above copyright notice, this
  list of conditions and the following disclaimer.

* Redistributions in binary form must reproduce the above copyright notice,
  this list of conditions and the following disclaimer in the documentation
  and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/

function levenshtein(__this, that, limit) {
  var thisLength = __this.length,
    thatLength = that.length,
    matrix = [];

  // If the limit is not defined it will be calculate from this and that args.
  limit = (limit || (thatLength > thisLength ? thatLength : thisLength)) + 1;

  for (var i = 0; i  (limit || 100)) {
    return prepare(limit || 100);
  }
  if (thisLength === 0) {
    return prepare(thatLength);
  }
  if (thatLength === 0) {
    return prepare(thisLength);
  }

  // Calculate matrix.
  var j, this_i, that_j, cost, min, t;
  for (i = 1; i  4) return prepare(thisLength);

      that_j = that[j - 1];
      cost = this_i === that_j ? 0 : 1; // Step 5
      // Calculate the minimum (much faster than Math.min(...)).
      min = matrix[i - 1][j] + 1; // Devarion.
      if ((t = matrix[i][j - 1] + 1)  1 &&
        j > 1 &&
        this_i === that[j - 2] &&
        __this[i - 2] === that_j &&
        (t = matrix[i - 2][j - 2] + cost) 
Now, this is the R code: 
library(DT)

dtable ').on('input', function() { fn(that, this) });",
              "  if (preDefined !== null) {",
              "     $(el).val(preDefined[0]);",
              "  }",
              "  return el;",
              "}"
            ),
            inputValue = JS(
              "function (el) {",
              "  return [$(el[0]).val()];",
              "}"
            ),
            isInputValid = JS(
              "function (el, that) {",
              "  return $(el[0]).val().length !== 0;",
              "}"
            ),
            search = JS(
              "function (value, pattern) {",
              "  var fuzzy = levenshtein(value, pattern[0]);",
              "  return fuzzy.similarity > 0.25;",
              "}"
            )
          )
        )
      )
    )
  )
)

path 
The similarity threshold must be selected. Here I take 0.25: 
return fuzzy.similarity > 0.25;



edit
To use in Shiny, use server=FALSE:
renderDT({
  dtable
}, server = FALSE)