I have not personally used 2Checkout but would like to share with you what I have read about it. The website itself did not specify processing time for application. I have summarized the application procedure below.
First you sign up with 2Checkout by submitting an application. Your application including your website will be reviewed by their underwriting team where they will request you to submit some documents like government issued id, proof of billing address, additional proof of address if address displayed from the website is not the same as billing address. They may also look at your products and services, marketing techniques, pricing and order process including refund and private policy. You may also get a call for validation purposes. After the application is approved and your site is up and running, you can start selling with 2Checkout.
There are pros and cons in using the services of 2checkout. Paypal seems to be the popular choice. You may check 2 articles or reviews made sometime May, 2017 via websites cardpaymentoptions and merchantmavericks, which will give you a total glimpse on whether to pursue with this payment method or not. Both contents feature feedbacks from actual users.
