Я нашел способ сделать это, используя метод GetSchema класса .NET Connection.
Я написал метод, который возвращает имена столбцов для конкретной таблицы.
Private Function GetColumnNamesInTable(ByVal connectionString As String, ByVal tableName As String) As List(Of String)
Dim connection As OleDbConnection = New OleDbConnection(connectionString)
Dim restrictions As String() = New String() {Nothing, Nothing, tableName, Nothing}
connection.Open()
Dim dataTable As DataTable = connection.GetSchema("Columns", restrictions)
connection.Close()
Dim returnList As List(Of String) = New List(Of String)
For Each dataRow As DataRow In dataTable.Rows
returnList.Add(dataRow("Column_Name"))
Next
Return returnList
End Function