I got an EIN so I could give that out rather than my SS#. Other than that I'm not aware of any benefit, unless Sunshine is correct about asset protection.
On the other hand, some shoppers have said in past threads on this topic that local authorities learned they had an EIN and required a business license. That's a local issue that doesn't seem to be a problem in most jurisdictions.
IANAL nor am I a CPA or an EA, and this is not legal or tax advice. However, YMMV.
In many states a single member LLC/disregarded entity LLC is much easier to pierce the veil of than that of a multi-member LLC because there's no other members' interests to protect (and there are ways around that, but then it makes the tax situation more complex). Also, if you form an LLC, you need to keep up with its legal and tax requirements which vary from state to state.
The other issue about EIN vs SSN is whether your secret shopping is considered a business or a hobby by the IRS. In a hobby you are entitled to all the business expenses but you cannot claim a net loss in your activities. Secret shopping is *not* my main source of income, so for me, I treat it as a hobby.
For example, when I go on a meal shop invariably I spend more than the fee pays, but that difference is not a tax loss for me. If my mileage exceeds my pay, I'm not entitled to claim a net loss against the fee.
Again, this is only informational in nature and may not apply to your situation. Get competent legal and tax advice if you feel that it's needed.