For customers receiving payment, will I need to collect their SSN and
issue them 1099s for income more than $600 in the year?
Yes. But depends on how your business is set up, it may be that you need to issue form 1099-K, not 1099-MISC, and then the thresholds are different. You'll have to talk to a (good, don't get cheap on this) CPA licensed in your State. You might want to have that consultation before you set your business up, so that you could decide which setup would be better for you,
I am assuming I can collect that information at years end so I don't
have to permanently store SSNs and use a 1099 service provider to
Technically, you have to have this information before you remit any payments. Any payments, so if someone wants to be paid immediately, and not at the year end - you have to have the information (form W9/W8-ECI) from them. Since the amounts may accumulate (someone gets paid $100 every month), you don't know ahead of time if you're going to issue 1099 for them or not. So you will have to keep that information, no way around it (you will also have to keep the copies of the 1099 forms, as well, so be prepared for the sensitive data being recorded and kept at your business).
If any major legal liabilities jump out at you regardless if you can
answer the original question please still chime in.
Since you're dealing with collecting and remitting payments, you might be classified as a merchant/payment operator. There are specific rules both on State level and on the Federal level, and some additional licenses might be required. Talk to a lawyer licensed in your state. Again, don't try to save on the consultations. It may cost you dearly later.
Last but not least: INSURANCE and liability protection. Start with the lawyer, continue with the insurance adviser.