MFMailComposeViewControllerを使ってメールを作成します。

まずはMessageUIをインポートします。

import MessageUI

デリゲートを追加します。

class sample: UIViewController, MFMailComposeViewControllerDelegate {

func mail(subject: String, mailbody: String) {
    let composeVC = MFMailComposeViewController()
    composeVC.mailComposeDelegate = self

    composeVC.setToRecipients(["送信先メールアドレス"])

    composeVC.setSubject(subject)

    composeVC.setMessageBody(mailbody, isHTML: false)

    let layer = UIApplication.shared.keyWindow?.layer
    let scale = UIScreen.main.scale
    UIGraphicsBeginImageContextWithOptions((layer?.frame.size)!, false, scale)

    layer?.render(in: UIGraphicsGetCurrentContext()!)
    let screenshot = UIGraphicsGetImageFromCurrentImageContext()
    let image = UIImageJPEGRepresentation(screenshot!, 1.0)
    UIGraphicsEndImageContext()

    composeVC.addAttachmentData(image!, mimeType: "image/png", fileName: "screenshot.png")

    self.present(composeVC, animated: true, completion: nil)
}

デリゲートで必要なメソッド
メール送信やキャンセルの操作を受けて実行されます。

func mailComposeController(_ controller: MFMailComposeViewController, didFinishWith result: MFMailComposeResult, error: Error?) {
    controller.dismiss(animated: true, completion: nil)

    switch result {
    case .sent: //送信
            break
    case .saved: //保存
            break
    case .failed: //失敗
            break
    case .cancelled: //キャンセル
            break
    default:
        break
    }
}

Xcode: 8.3.2
Swift: 3.1
OS: Sierra 10.12