There is ALWAYS a risk in giving out private information like driver license information and social security numbers on the net. Is the form secure? I cannot think it would not be. But if not, the information can be intercepted by third party.
Also, you are in the hands of Bittrex. I am sure they have some routines to handle personal information to comply with regulations but still. All it takes is one rotten employee to abuse your information. Not saying in will happen, but nothing is 100% secure.
But if you are going to transfer your hard-earned money to them anyway then why not? I just do it. someone is bound to make a mistake so that my personal information will be all over the internet at some point