Saving datagridview data at form close to embedded file

You don't want to change an embedded file if you can possibly avoid it: it means altering an executable file which is the kind of activity that virus checkers do look for...
Instead, store the info in a standard external file (see here for suggestions as to where to store it: Where should I store my data?[^])

An SqLite DB should work ok, but if you want to just save and restore the data via a DataTable, then the DataSet has methods to read and write XML files directly:
Easy write:
Create your data in a DataTable, then:

DataSet ds = new DataSet();
ds.Tables.Add(dt);
ds.WriteXml(@"D:\Temp\td.xml");

Easy Read:

DataSet ds = new DataSet();
ds.ReadXml(@"D:\Temp\td.xml");
dt = ds.Tables[0];

You can use DataTable.ReadXml instead, but it doesn't support schema inference, DataSet.ReadXml does.