I would argue that the e-mail contact form is a thing of the past. In fact, I find it very annoying when I encounter websites – in particular commercial websites – that do not display their e-mail addresses and other contact information properly, but instead rely on a web based e-mail form through which existing and potentially new customers can contact them. And why do I think so?

The e-mail contact form - a thing of the past?
Control
First of all, I like to be in control, and with e-mail forms I seldom feel that I am in control. How can I be sure the e-mail for works as intended? How can I be sure the e-mail was actually received?
When I send my e-mails through my regular e-mail client (Outlook, Gmail, etc.) I have a copy in my “Sent Items” folder – and that leaves me in control. I know what e-mails I have sent, and if necessary, I can refer to them later on. With web forms, e-mails are sent through a “black hole” and often never seen again. Which leaves me to talk about confirmation.
Confirmation
Sending an e-mail through an e-mail form is like shooting a bullet straight up in the air – you’ll never know where it is going to land – and if it ever landed. Some e-mail forms do offer the option to have a copy of the message sent to your own e-mail address – however – it is not a confirmation that the message was ever received by the recipient.
Referring to my previous point – if I send my e-mail from my regular e-mail client, I do not necessarily get a confirmation of every delivered e-mail, but I least I (usually) get an error message if e-mails I send cannot be successfully delivered.
(Lack of) Professionalism
If you as a company do not want to reveal you contact e-mail address to visitors of your website, I start wondering about why you need to be so secret? What do you need to hide from the public? Is spam a problem, and you cannot afford to deal with it, well, in my view that says a lot about you as a company – that you are not professional. Most companies find their way to fight spam – and so should you. Don’t blame your customers and leave them to pay for your lack of professionalism.
My recommendation
I recommend to always display your contact e-mail address in the open, e.g. in the footer of your webpage. E-mail forms are just an annoyance, and I would recommend to have them removed from you website.