There are a large number of choices for platforms to support the issuing of badges. Some institutions have built their own apps and other contract with commercial providers and there are also open source and free options. The larger message I would like to be sure is heard is to check that your badge platform issues badges compliant with the Open Badge Standard. Badges compliant with this standard are portable to other systems in the event of failure of the company or platform. The data systems that are referenced by your badges are a different question and one worthy of some serious thought on your part. IMS Global is the organization that now supports and maintains the Open Badge Standard.
Free
Badgr - Open Badge Compliant
Badgr is a simple and accessible badge issuing platform for open badge compliant badges. If it meets your needs you will discover that you are also supporting the work of Nate Otto who is one of those who was there when badges were born. Without his unwavering commitment to the cause it is not clear where badges would be today though he has not labored alone.
Commercial (Domestic)
Credly - Open Badge Compliant
I have no direct experience since I've been building badges by hand for now. Good experiences from those who use this platform.
Acclaim - Open Badge Compliant
I have no direct experience since I've been building badges by hand for now. Good experiences from those who use this platform.
Canvas - Open Badge Compliant
Canvas supports the issuing of badges though I am not particularly aware since the LMS I operate under is Blackboard which is not Open Badge Compliant (a deal breaker for me)
Blackboard -- NOT compliant
Don't bother! You do your students a disservice by issuing badges that are not portable
Commercial (International)
Open Badge Factory - Open Badge Compliant
Discendium operates the Open Badge Factory and Eric Rousseau is it's principle. He is an ardent badge evangelist and supporter whom you will meet at almost any badge event.
In House (eventual availability unclear)
Badge issuers listed here have built their own badge platforms and may or may not ever make those systems available to the broader community.
Colorado State University
LRNG