API help needed for login on desktop program with remote Instagram Accounts

Hi, I am currently using a desktop program for my business. The program works with social media and it requires a login via twitter or instagram. I want to start doing using this program for my customers, but the problem is some of my customers live far away and I don't want them telling sensitive information (their password) However, we are all fine with me receiving access if they were able to "remotely login" or "remotely grant access" So I need you to create an ability so that it results in them being logged into their account on my computer program. I'm fine 'presenting' it however you think best: An app, plug-in, or feature on my site. I was informed about API being of help but I don't know much more about it. Also, I was told to look for someone proficient in computer app programing. Please reply back with how you plan on accomplishing this task. As well as showing me any proof of concept.