How do I figure my Roth contributions if they were made randomly prior to 2015?

Unfortunately, it is your responsibility to keep track.  The IRS does not have to give you credit for any tax break (like tax-free withdrawals from a Roth IRA) unless you can prove it with your own reliable records.

You don't need to know your contributions if you are over age 59-1/2, you only need to know if you are younger.  So if older, make your best guess (I don't even know why the program asks).  If you are younger, and you need to know your contributions because you want to make a withdrawal and you need to know if part of the withdrawal is taxable, you will have to do the best you can with gathering your own personal records.  You can ask the IRA custodian if they have records of your contributions.  The IRS will have copies of the form 5498s that were sent to you each year, but the IRS copies only go back 10 years.