Better method for updating data table
The only part that executes slowly would be the for and the nested
  for. Is there possibly a better way that I can code this?


I take you ask us to optimize the nested loops:

The most obvious problem, as @Antonio also noted, is the O(n^2) filtering in the nested fors. Also as he noted, you need some kind of data structure with O(1) Contains method. However since you do not use other fields of from the rows of ModelResults, you only need a HashSet of ["Request ID"]s. 


  I can't imagine this is the optimal coding for data table update
  operations, maybe though? Is this a possibility in LINQ (I'm
  completely new to LINQ)?


I take you are asking for a more compact, more LINQy way of updating a collection, in this case a DataTable:

LINQ is a query language and a LINQ query does not modify underlying IEnumerable. (In most cases you would select a modified copy of the underlying IEnumerable, instead.)

However, you can use List.ForEach on the result of such a query, as a more compact replacement of for loop.

Applying two suggestions above, we get:

var collectedRequestIds = new HashSet&lt;string&gt;(
    modelResults.AsEnumerable().Select(row =&gt; row["Request ID"].ToString()));

modelData.AsEnumerable()
    .Where(collectedRequestIds.Contains(row =&gt; row["Request ID"].ToString())))
    .ToList().ForEach(row =&gt; row["Collected"] = "1");