You can better use UserVoice contact form. Uservoice will do two things
It will create new ticket inside Uservoice system.
It also send a mail to you.
It will be end to end solutions for customer support. However if you send mail through some services, you have to communicate to your customer for support. Uservoice will designed and built for customer support only.
I know, you have a questions that you want to use your own custom form, not the one provided as uservoice widget. Yes, you can do it by
https://gist.github.com/cypriss/2032215