There is no limitation for a regular SELECT column names FROM SomeTable.
Unless there is a really solid reason for using .mdb move to .accdb and try with similar code to the following. See if that works or not before using a DataAdatper. Also double check your connection string is pointing to the correct database.
public static DataTable GetAll()
{
DataTable table = new DataTable();
using OleDbConnection cn = new() { ConnectionString = "TODO"};
using OleDbCommand cmd = new() { Connection = cn };
cmd.CommandText = "SELECT Id, FirstName, LastName FROM Person";
cn.Open();
table.Load(cmd.ExecuteReader());
return table;
}